Wave interaction with permeable structures - Oregon State ...

AN ABSTRACT OF THE THESIS OF

Chung-Pan Lee for the degree of Doctor of Philosophy in

Civil Engineering presented on August 11, 1987.

Title: Wave Interaction with Permeable Structures

Abstract approved:

Redacted for PirivacyCharles K. Sollitt

A theory is developed to provide an analytical solution to an

unsteady flow field which is partially occupied by a porous struc-

ture. The flow is induced by a small amplitude incident wave

train. The porous structure may contain multi-layer anisotropic but

homogeneous media. Three typical porous structures are investi-

gated: a seawall with toe protection, a rubble-mound breakwater,

and a caisson on a rubble foundation. Theoretically, however, any

two-dimensional porous structure with an arbitrary geometry can be

treated by this analytical procedure.

Resistance forces in the porous structures are modeled as

inertia forces, skin friction drag, and form drag. Form drag is

empirically nonlinear and is replaced by a linear drag term utilizing

Lorentz's condition of equivalent work. The periodic small motion

can then be shown to be irrotational and a single-valued velocity

potential is defined. The velocity potential satisfies a partial

differential equation which reduces to the Laplace equation when the

porous structures are isotropic.

The flow domain of a porous structure with inclined boundaries

is first partitioned and approximated by a group of rectangular,

layered sub-domains. An eigenseries representation of linear wave

theory in each sub-domain is then solved from the imposed boundary

value problem by the method of separation of variables. The

kinematic and the dynamic boundary conditions on the boundary between

any two adjacent sub-domains and on the free surface are matched.

The kinematic boundary condition on any impermeable boundary is

satisfied. The solution in the sub-domain with an open boundary at

infinity also satisfies Sommerfeld's radiation condition.

A large scale experiment of a seawall with toe protection has

been conducted to validate the theory. Measurements included the

pressure distribution above and within the structure and the incident

and reflected wave characteristics. Theoretical and experimental

dynamic pressures in the toe compare very well. In addition, theo-

retical and experimental reflection coefficients follow the same

trend of decreasing magnitude with increasing wave number and wave

steepness.

Wave Interaction with Permeable Structures

by

ChungPan Lee

A THESIS

submitted to

Oregon State University

in partial fulfillment ofthe requirements for the

degree of

Doctor of Philosophy

Completed August 11, 1987

Commencement June 1988

APPROVED:

Redacted for PrivacyProfessor of Civil Engineering in charge of major

(Redacted for Privacy

Head of of Cliv, IEngineering

Redacted for Privacy

Dean of Graduat

(7shoo]

Date thesis is presented August 11, 1987

Typed by Peggy Offutt for Chung-Pan Lee

ACKNOWLEDGEMENTS

I would like to express my most grateful acknowledgement to my

advisor, Dr. Charles K. Sollitt, for his advice, concern, considera-

tion, and full support thoughout the three years I have worked for

him.

Gratitude is also given to Mr. David R. Standley, research

scientist, and Mr. Terry L. Dibble, research engineer, both of the

O.H. Hinsdale Wave Research Laboratory, and Mr. Tsung-Muh Chen, Ocean

Engineering graduate student, for sharing their knowledge and con-

tributing significantly to the experiments and the data analysis.

Appreciation is further extended to Ms. Peggy Offutt for her patience

and skill in typing the manuscript and the extra effort required to

neatly and accurately reproduce the equations. Thanks is also

expressed to my family for their support and encouragement.

This research was sponsored by NOAA Office of Sea Grant, Depart-

ment of Commerce, under Grant No. NA86AA-D-SG095 (Project No.

R/CE-18). The U.S. Government is authorized to produce and distrib-

ute reprints for governmental purposes, notwithstanding any copyright

notation that may appear hereon.

TABLE OF CONTENTS

1. INTRODUCTION

1.1 Purpose of the Study

1.2. Literature Review

Page

1

1

3

1.2.1 Common concerns of porous structures 3

1.2.2 Porous structure stability 7

1.2.3 Wave reflection and transmission 8

1.3 Need for Additional Research 10

1.4 Scope 11

2. POTENTIAL THEORY 13

2.1 Introduction 13

2.2 Equations of Motion 14

2.3 Implications of Anisotropic, Homogeneous Media 16

3. BOUNDARY CONDITIONS 19

3.1 Introduction 19

3.2 Seawalls with Toe Protection 25

3.3 Rubble-Mound Breakwaters 28

3.4 Caisson Structures on a Rubble Foundation 31

4. ANALYTICAL SOLUTIONS 34

4.1 Introduction 34

4.2 Separable Equations of Motion 37

4.3 Determining the Unknown Coefficients of theZ-Dependent Term And the Dispersion Equation 39

4.3.1 Columns with a free surface 39

4.3.2 A column with an impermeable upper boundary 45

4.4 Determining the Relationship Between Eigenvalues inthe Same Column 47

4.5 Determining the Unknown Coefficients of theX-Dependent Term 50

4.5.1 Relationship between the unknown coefficientsof the x-dependent term in the same column.... 53

4.5.2 Specific conditions for seawalls with toeprotection 56

4.5.3 Specific conditions for rubble-moundbreakwaters 66

4.5.4 Specific conditions for caisson structures ona rubble foundation 69

5. ANALYTICAL SOLUTION BEHAVIOR 73

5.1 Material Properties 73

5.2 Computation Procedures 74

5.3 Theoretical Results 79

6. EXPERIMENTAL STUDIES: A SEAWALL WITH TOE PROTECTION 99

6.1 Wave Testing Facilities 99

6.2 Test Procedures 102

6.3 Material Properties 106

7. COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS 110

7.1 Introduction 110

7.2 Comparison of Experimental and Theoretical Results 111

8. CONCLUSION 130

8.1 Summary 130

8.2 Theoretical Behavior 133

8.3 Comparison with Experiments for Seawall Toes 133

8.4 Future Investigation 134

9. REFERENCES 136

10. APPENDICES 138

LIST OF FIGURES

Figure Page

1.1(a) A seawall with toe protection 4

1.1(b) A rubble-mound breakwater 5

1.1(c) A caisson structure on a rubble foundation 6

3.1(a) Rectangular partitions of flow domains with inclinedboundaries of a seawall with toe protection 22

3.1(b) Rectangular partitions of flow domains with inclinedboundaries of a rubble-mound breakwater 23

3.1(c) Rectangular partitions of flow domains with inclinedboundaries of a caisson structure on a rubblefoundation 24

3.2 Boundary conditions on the horizontal boundaries ina column with the free surface 27

3.3 Boundary conditions on vertical permeable boundaries 29

3.4 Boundary conditions on an impermeable verticalboundary 30

3.5 Boundary conditions on the boundaries which compriseof both permeable and impermeable parts

3.6 Boundary conditions on the impermeable boundaries ofthe layer under the caisson

4.1 The construction of equations from boundaryconditions on vertical boundaries for the case of aseawall with toe protection

4.2 The construction of equations from boundaryconditions on vertical boundaries for the case of arubble-mound breakwater

32

33

58

68

4.3 The construction of equations from boundaryconditions on vertical boundaries for the case of acaisson structure on a rubble foundation 70

5.1 The flow chart of theoretical computation procedures.. 76

5.2 Definition sketch. Broken line is the originalinclined surface 80

5.3 Reflection coefficient dependence on linear dragcoefficients, where d = 0.5 h 81

5.4 Disturbed wave length dependence on linear dragcoefficients, where kh = 3.1 for T = 2 sec, kh a 1.0for T = 4 sec, and d = 0.5 h 83

5.5 Horizontal fluid particle velocity dependence onlinear drag coefficients, where d = 0.5 h 85

5.6 Reflection coefficient dependence on porosities,where d = 0.5 h 87

5.7 Disturbed wave length dependence on porosities, wherekh = 3.1 for T = 2 sec, kh = 1.0 for T = 4 sec, andd = 0.5 h 88

5.8 Horizontal fluid particle velocity dependence onporosities, where d = 0.5 h 89

5.9 Reflection coefficient dependence on virtual masscoefficients, where d = 0.5 h 91

5.10 Disturbed wave length dependence on virtual masscoefficients, where kh a 3.1 for T = 2 sec, kh - 1.0for T a 4 sec, and d = 0.5 h 92

5.11 Horizontal fluid particle velocity dependence onvirtual mass coefficients, where d = 0.5 h 93

5.12 Reflection coefficient dependence on toe geometry 95

5.13 Distrubed wave length dependence on toe geometry,where kh = 3.1 for T = 2 sec, kh a 1.0 for T = 4 sec.. 96

5.14 Horizontal fluid particle velocity dependence on toegeometry. At "Top." 97

5.15 Horizontal fluid particle velocity dependence on toegeometry. At Toe." 98

6.1 Wave testing facility 100

6.2 Location of the pressure transducers in the toe.Unit: ft 101

6.3 Pressure transducer mounting bracket 103

6.4 Size distribution of porous media. Curve 1 wasdetermined from the major and minor dimensions ofindividual rocks. Curve 2 was determined from theweight of individual rocks 108

7.1

7.2

7.3

7.4

7.5

7.6

7.7

Reflection coefficient






Nondimensional dynamictoe: a) comparison ofat numbered positions,bution at grid points,measured values (which

7.8 Nondimensional dynamictoe: a) comparison ofat numbered positions,bution at grid points,measured values (which





dependence on h/Lo 113



dependence on wave steepness..



pressure distribution in thepredicted and measured valuesb) predicted pressure distri-c) predicted isobars andhave three decimals) 120

pressure distribution in thepredicted and measured valuesb) predicted pressure distri-c) predicted isobars andhave three decimals) 121

pressure distrubtion in thepredicted and measured valuesb) predicted pressure distri-c) predicted isobars andhave three decimals) 122

pressure distrubtion in thepredicted and measured valuesb) predicted pressure distri-c) predicted isobars andhave three decimals)

pressure distrubtion in thepredicted and measured valuesb) predicted pressure distri-c) predicted isobars andhave three decimals)

123

124


Table

7.1

7.2

LIST OF TABLES

Stream Function Cases for h 12 Feet

Stream Function Cases for h 10 Feet

Page

112

112

WAVE INTERACTION WITH PERMEABLE STRUCTURES

1. INTRODUCTION

1.1 Purpose of The Study

In ocean engineering, porous structures such as rubble-mound

breakwaters have been widely constructed to provide safety for navi-

gation. Applications also include toe protection for seawalls and

caisson-type structures. Porous structures provide shelter from wave

attack by reflecting and dissipating incident wave energy. In

permeable breakwaters, part of the incident wave energy is trans-

mitted through the porous structure. The distribution of reflected,

transmitted, and dissipated wave energy is strongly affected by water

depth, wave properties such as wave period and wave height, and

structure properties. The major structure properties are geometry,

porosity, permeability, size distribution and shape function of the

components of the porous structures.

These factors also determine the stability of the structures,

reflection and/or transmission, wave runup and rundown, etc, under

wave attack in a given water depth. A study to optimize the cost of

constructing a secure and efficient structure is usually needed.

This requires an understanding and prediction of the flow inside and

outside the structure and the force distribution on the structure

under given wave conditions. Unfortunately, at the present time,

only very simplified structure geometries (e.g., crib-type break-

waters: Sollitt and Cross, 1972; P. L.-F. Liu, at al, 1986) can be

examined analytically. And most of the numerical methods have been

2

developed only for specific structures (Hannoura and McCorquodale,

1985; Kobayashi and Jacobs, May 1985; Kobayashi, Otta, and Roy,

1987). Therefore, many design problems concerning porous structures

are still solved by empirical equations such as the Hudson equation

or by experimental tests.

With the existence of both waves and porous structures in the

flow domain, the nonlinearity at the free surface and the boundaries

between different media is one of the major difficulties. The random

arrangement of the porous media precludes any precise description of

the rigid boundaries of the flow inside structures. Turbulence in

the flow field with high Reynolds number is another major unknown.

With all the difficulties mentioned and the associated uncertainties,

a solution to the problem is sought by combining approximate forms of

the basic fluid dynamic equations of motion with careful and precise

analysis.

In this study, nonlinearities at the boundaries are not included

in the solution. Linear waves are considered to give a first order

approximation. The randomness of the porous media suggests that only

average porous properties can be considered. The main problem of

turbulence is the lack of analytical stress-strain relations. An

empirical expression which is similar to the drag term in the Morison

equation is therefore used to represent unsteady resistance (Steimer

and Sollitt, 1978).

Homogeneous and isotropic porous media have been assumed in most

analyses of waves and porous media. However, many real porous struc-

tures can be considered neither as homogeneous nor isotropic. In

3

this study, anisotropic porous media are considered. Homogeneity

within sub-structures is still assumed. However, a nonhomogeneous

structure is approximated as an assembly of rectangular sub-

structures of unique yet homogeneous properties. The purpose of this

study then is to develop an analytical solution for the flow field

with a porous structure which may contain multi-layer anisotropic but

homogeneous media and may have inclined surfaces. Three typical

porous structures are considered in the study. They are a seawall

with a porous toe, a rubble-mound breakwater, and a caisson structure

on a rubble foundation, as shown in Figure 1.1. However, the pro-

cedure developed here to solve the flow fields can be applied to any

two dimensional flow field with a porous structure of any geometry.

1.2 Literature Review

1.2.1 Common concerns of porous structures

The common concerns related to porous structures are wave runup

and rundown, overtopping, wave reflection and/or transmission, and

structure stability. Runup is the vertical distance above the still

water level caused by a wave rising on some prescribed surface. Run-

down is the vertical distance between the still water level and the

minimum elevation attained by a wave on a specified surface. Runup

and rundown on porous structures are usually studied by physical

model tests (e.g. Sollitt and DeBok, 1976). Sometimes, the porous

structures are further simplified to be impermeable but with rough

surfaces (e.g. Gopalakrishnan and Tung, 1980; Hannoura and

McCorquodale, 1985; Kobayashi and Jacobs, May 1985; Kobayashi, Otta,

and Roy, 1987). Whether overtopping will occur depends on the height

Figure 1.1(a). A seawall with toe protection.

.P-

Figure 1.1(b). A rubble-mound breakerwater.

U'

Figure 1.1(c). A caisson structure on a rubble foundation.

Cr)

7

of the crest of the structures relative to the height of the wave

runup. Breaking waves may also occur outside the structure and cause

tremendous impact forces. Reliable data about impact forces are

obtained from experiments. Wave reflection and/or transmission, and

structure stability are reviewed in more detail in the following

sections.

1.2.2 Porous structure stability

A porous structure's stability is often quantified by the weight

of its armor units. This is because weight resists the destabilizing

forces of lift, drag, and inertia. The armor unit weight for a

porous structure is usually determined by empirical equations. One

of the most popular equations is Hudson's formula. The equation is

simple and easy to use; however, because of its simplification, it

does not include some important effects. Some of the effects are the

material properties, the position where waves are breaking, wave

periods, local kinematics, local pressure, etc. Some improved equa-

tions presented by Hedar in 1986 are provided to include some of

these effects for the determination of the weight of armor units.

Physical model tests are often prescribed for unique armor unit

applications. The interpretation of a model test with surface waves

is usually done by following the Froude law, which simulates gravita-

tional forces. The model simulation, however, violates Reynolds law,

which simulates viscous forces. In an energy dissipation region,

such as in a porous structure, viscous effects are significant and

can not be neglected. The violation results in the so-called scale

effects. By comparing large and small scale model tests, scale

8

effects reveal that the relative increase in drag forces at lower

Reynolds numbers is shown to decrease the stability and runup in

small scale models (Sollitt and DeBok, 1976).

As an alternative to physical modeling, the stability of a

porous structure may be simulated by simplified mathematical models

which retain the roughness of the structure's slope but assume imper-

meable boundaries (e.g. Kobayashi and Jacobs, Sep, 1985). This

assumption ignores the dynamic behavior of interstitial flow and the

corresponding destabilizing forces on armor units.

Hannoura and McCorquodale (1985) developed a numerical model to

study the wave induced flow in a rubble-mound breakwater and, con-

sequently, provided a criterion for the structure's stability. The

numerical model combined a finite difference method to determine the

internal water levels and a finite element method to solve the two

dimensional flow within the structure. In their study, shallow water

waves were assumed. The pressure distribution on the surface of the

structure was required as input data and was determined in advance by

experiments. The model has been applied to the Port Sines breakwater

that failed recently. Their result showed a lower factor of safety

than the traditional analysis. Virtual drags were included in the

analysis. Internal wave breaking and the entrainment of air near the

interface were empirically taken into consideration as well.

1.2.3 Wave reflection and transmission

Sollitt and DeBok's work in 1976 included the measurement of

wave reflection. Their results show that reflection increases with

decreasing wave steepness and increasing model size. This is because

9

the surface drag on the breakwater surface is a nonconservative force

which increases with the square of local fluid particle velocities,

and additionally, fluid particle velocities increase with increased

wave steepness. Thus, a steeper wave dissipates more energy than a

less steep wave. Similarly, the low Reynolds number flow in a small

model causes proportionately higher drag and reduces reflection.

In 1972, a theory was derived by Sollitt and Cross to analyti-

cally predict the wave reflection and transmission caused by a perme-

able crib-type breakwater. The theory included the virtual drag and

the viscosity caused friction drag and form drag in the structure.

The form drag was found empirically to be proportional to the square

of local fluid particle velocities. The nonlinearity of the form

drag was resolved by applying Lorentz's condition of equivalent work

and replaced by a linear drag which consumes the same energy as non-

linear drag in one wave period. The flow in the structure induced by

small amplitude waves was then shown to be irrotational, and a poten-

tial theory was applied to solve the imposed boundary value problem

analytically.

For a conventional trapezoidal breakwater, an equivalent rectan-

gular breakwater was first constructed such that its submerged volume

was the same as that of the original breakwater. The theory of

Sollitt and Cross (1972) can then be applied to solve the velocity

potentials inside and outside the structure. The velocity potentials

are used to calculate all the flow properties such as reflected and

transmitted wave amplitudes, and dynamic wave pressures by

Bernoulli's equation. Fluid particle velocities can be calculated

10

through the definition of the velocity potentials. Experiments have

been done by Sollitt and Cross (1972) to verify the theory. Their

results show that the transmission coefficient decreases with

increasing wave steepness, and reflection coefficient is relatively

insensitive to changes in wave steepness. The study also empirically

included energy dissipation due to breaking waves on the surface of

the breakwater.

Madsen and White (1976) developed a similar model for long waves

which solves the flow field within a conventional trapezoidal break-

water. However, their model still requires the reduction of the

trapezoidal breakwater to a hydraulically equivalent rectangular

breakwater. The equivalent breakwater geometry was scaled to allow

the same hydraulic discharge as the original breakwater.

Recently, Liu, et al (1986) also developed an analytical model

to solve the flow field inside and outside a crib-type breakwater.

The energy dissipation in the porous structure was, however, modeled

by the mild-slope equation, describing the propagation of a linear

periodic wave train, with damping used by Booij (Liu, et al.,

1986). In the equation, the rate of energy dissipation is equal to

the divergence of wave power. This model has been developed

numerically to solve a three dimensional problem.

1.3 Need for Additional Research

In previous studies concerning wave interaction with permeable

structures, analytical models can only be applied to crib-type break-

waters which are homogeneous and isotropic. The porous structure

must extend to the free surface. For a layered trapezoidal break-

11

water, an equivalent homogeneous rectangular breakwater must be

constructed in order to apply the models. For a submerged permeable

structure, previous analytical models can not be applied because the

porous structure only occupies a portion of the water column. They

also can not be applied to a flow domain where part of the water

column is impermeable, e.g., when a caisson is placed on a rubble

foundation.

Nonlinearity is another unresolved problem in previous analyti-

cal models. Basically, the difficulty occurring in analytical models

is mainly due to the nonlinearity of the boundary conditions on

inclined boundaries and on the free surface. Rotational or turbulent

flow inside porous structures also raises another difficulty.

1.4 Scope

The scope of this study is to provide an analytical model which

solves a boundary value problem containing layered porous structures

with inclined boundaries. The structure may be submerged or semi-

submerged. Porous media are still assumed to be homogeneous, but

they are considered to be anisotropic (i.e., not isotropic).

Nonlinearity is not included in the study. Linear amplitude

incident waves are required and the responding motion may then be

linearized to give a first order approximation.

Three typical porous structures are investigated. They include

a seawall with toe protection, a rubble-mound breakwater, and a cais-

son on a rubble foundation. However, the analytical procedure

developed in this study has been generalized such that it can be

12

applied to a flow field containing any twodimensional porous struc

ture with an arbitrary geometry.

The results of this study are to be used to provide a quantita

tive description of the kinematic environment on the slope of a

variety of porous structures. This kinematic description is to be

combined with a Morison equation stability model, proposed by Chen

(1987), to yield a rational predictive model for armor stability.

13

2. POTENTIAL THEORY

2.1 Introduction

Flows in anisotropic porous structures induced by a linear wave

train follow a modified form of the Navier-Stokes equations (Sollitt

and Cross, 1972). In the equations, the resistance terms include an

inertia force which is proportional to fluid particle accelerations,

a skin friction drag which is proportional to fluid particle veloci-

ties, and a nonlinear form drag which is proportional to the square

of fluid particle velocities. The nonlinear form drag can be

replaced by a linear drag from Lorentz's condition of equivalent work

(ibid). The condition requires that energy dissipation is the same

in both the linear and nonlinear drag models during one wave

period. For a periodic motion, a modified flow field can be defined

and proved to be irrotational. That is, the corresponding vorticity

can be shown to be independent of time, and if the flow is initially

irrotational it will remain irrotational. For an irrotational flow,

a single-valued velocity potential can be shown to exist. The conti-

nuity equation in incompressible fluid shows that, for homogeneous

media, the velocity potential satisfies a partial differential equa-

tion which reduces to the Laplace equation for isotropic porous

structures (ibid). For convenience, the differential equation is

called the modified Laplace equation hereafter.

From the definition of the velocity potential, the velocity com-

ponent in each direction can be resolved once the velocity potential

is solved. Also, the pressure can be found by applying Bernoulli's

equation, derived from the integrated equations of motion. Thus,

14

instead of solving four unknowns (in three dimensions, three velocity

components, and the pressure) in four equations (three equations of

motion and the continuity equation) in an incompressible fluid, only

one unknown (the velocity potential) needs to be solved from one

equation (the modified Laplace equation) with appropriate boundary

conditions.

2.2 Equations of Motion

Consider a flow field within a multi-layer anisotropic but

homogeneous porous structure under the action of a small amplitude

incident wave train in constant water depth. The fluid is assumed to

be incompressible. The equations of motion of the flow can be

written as (Sollitt and Cross, 1972)

du.....J_

.1_ (resistance forces/unitdt p x.

Jmass) in the xj direction

(1)

where j = 1,2 in a two dimensional flow field, p is the water den-

sity, g is gravitational acceleration, xl - x, x2 = z where the ori-

gin of the coordinate system is at the still water level, ul = u,

u2 = w, and p is the pressure. In general, the resistance forces

include (1) skin friction drag in laminar flow and form drag in tur-

bulent flow and (2) the virtual force which is due to the relative

acceleration between the flow and an obstacle in the flow. The

resistance forces vanish outside porous structures where an inviscid

fluid is assumed.

Empirically, skin friction drag is proportional to uj, and the

form drag is proportional to u.J luJ d. This was established by Ward

15

(1964) for large porous media in steady flows. The virtual force is

proportional to dui /dt. Thus, Eq. (1) can be rewritten as

dui du1 a

dt-- (P + p gZ ) -.. a 0 11 I II I -$

dt(2)

where Bkj' k = 1,2,3, are the proportional constants which can be

determined empirically. The form drag and the virtual force are

similar to those in the Morison equation (Steimer and Sollitt,

1978). The viscous drag forces dissipate energy. However, the vir

tual force (also called inertia drag) can be considered to contribute

to the kinetic energy of an added mass and does not dissipate energy.

According to Lorentz's condition of equivalent work, a nonlinear

mechanism can be replaced by a linear mechanism if both consume the

same energy in one wave period (Sollitt and Cross, 1972). Then a

linear drag coefficient fj can be defined as

t +T t +Taf f ° au.)u dtdV = f f

ljuj+ 0

2juj

luj 1 )ujdtdV

(3)V t J 3 J

°

v t0 0

where a =2z/T, T is the wave period, to is any time, and V is the

flow domain considered. Equation (2) can then be written as

du.

(p + Pgz) f.auSj dt j

where EL =, (1+03j

) is called the virtual mass coefficient.

(4)

2.3 Implications of Anisotropic, Homogeneous Media

For small motions, the convective terms in Eq. (4) can be

neglected, and Eq. (4) reduces to

1 9a(f. iS.)u

1 jP(p + pgz)

J

16

(5)

foraperiodicmotion.uj a exp( iot), i = ri. From Eq. (5), a modi

fied velocity utc can be defined asJ

f iS

ui I )* ( J il u. (6)J

Then Eq. (5) can be rewritten as

1

axiou*

p(p + pgz)

and, for the periodic motion, Eq. (7) can also be rewritten as

al

at p(p + pgz)

3x.

(7)

(7a)

which is the Euler equation of the modified velocity. It should be

noted that pressures are the only surface forces in the Euler equa

tion. Taking the curl of Eq. (7) and Eq. (7a) gives

and

x u* = 0

a x 11*) = oat

(8)

(8a)

Equation (8) shows that the modified velocity is irrotational, and

Eq. (8a) shows that its vorticity is stationary or independent of

time in the linear approximation. Thus it is concluded that the flow

is always irrotational if it is initially irrotational. In an

irrotational flow, there exists a single-valued velocity potential

which can be defined as

asut3 axe

Combining Eq. (6) and Eq. (9) gives

u (-

)

axesi

17

(9)

(10)

Substitute Eq. (9) into Eq. (7a) and integrate in space to obtain the

Bernoulli equation

_it + .2 + gz = 03t p

where the time-dependent integration constant has been incorporated

into t. The continuity equation of an incompressible fluid is given

as

2 au.

-57j-P o

for a two dimensional flow. Substituting Eq. (10) into Eq. (12)

gives the modified Laplace equation

2

Li[(f - ]0

j.I ji 2Sj ax.

(12)

(13)

for homogeneous structures. This is a differential equation govern-

ing periodic small motions in an anisotropic but homogeneous porous

medium. After the velocity potential is solved, the velocity compo-

nents and the pressure can be found. from Eq. (10) and Eq. (11),

respectively.

Note that Eq. (10) and Eq. (13) reduce to

and

ILu .

ax

18

(10a)

72 + 0 (13a)

respectively, in clear water where an inviscid fluid is assumed.

19

3. BOUNDARY CONDITIONS

3.1 Introduction

Physical constraints impose the requirement that the velocity

potential in each porous medium subdomain matches kinematic and

dynamic boundary conditions on the boundaries separating adjacent

media domains. For potential flow in clear water, a dynamic boundary

condition (DEC) requires that the pressure is continuous across the

boundary separating two flow fields. This can be derived by applying

Newton's second law to a control volume which encloses the boundary

and then shrinking the control volume to the boundary itself where

pressures are the only surface forces. For a boundary separating two

immiscible fluids, such as the free surface, surface tension may be

important and needs to be included. For a boundary separating two

flow fields containing the same fluid, the boundary condition can

also be considered as a consequence of Newton's third law which

states that the reaction on one side of the boundary is equal to the

action on the other side of the boundary. Again, pressures are the

only forces on the boundary in potential flows. In addition, a kine

matic boundary condition (KBC) requires that the velocity normal to

the boundary is continuous across the boundary. This is the conse

quence of the conservation of mass.

These conditions in potential flows are different from those in

real fluids where stress and velocity rather than pressure (normal

stress) and normal velocity are required to be continuous across a

boundary. The difference in the dynamic boundary conditions is

obvious because shear stresses as well as normal stresses exist in

20

real fluids. However, the difference in the kinematic boundary con-

ditions is caused by the fact that real flows are observed to not

slip when they contact rigid boundaries. Consequently, even though

conservation of mass has been satisfied if the normal velocities are

continuous across a boundary, the continuity of tangential velocities

across the boundary is further required to insure the no-slip bound-

ary condition in real fluids.

When flux is defined as the flow rate through a surface, the KBC

at a fixed boundary for potential flow in porous media can be gen-

eralized as "the flux normal to a boundary is continuous across the

boundary." This continuity expression is equivalent to the require-

ment of the conservation of mass. In porous media, the generalized

KBC then requires that the product of the porosity and the velocity

normal to a boundary is continuous across the boundary. This gen-

eralized KBC reduces to the original KBC in clear water when the

porosity becomes unity. On a fixed impermeable boundary, this simply

means that no flow penetrates the boundary and the normal velocity

vanishes.

A porous structure such as a rubble-mound breakwater usually has

inclined surfaces. This raises a mathematical difficulty because the

boundaries of the flow domain are not parallel to each other. One of

the mathematical techniques, which may be applied to solve the

Laplace equation, is the method of separation of variables. The

Laplace equation, which is linear, always has a solution whose

variables are separable. However, this solution may not (and usually

will not) satisfy the imposed boundary conditions on boundaries which

21

are not parallel to any axis of the specified coordinate systems.

This is because the boundary conditions on such boundaries are

usually variable dependent and not separable. In a two dimensional

problem, a rectangular domain is one in which a rectangular

coordinate system can be set such that every boundary of the domain

is parallel to one of the axes. This suggests a procedure to solve

the flow domain with inclined boundaries by partitioning the original

flow domain into several rectangular sub-domains with multiple layers

in the same column. The rectangular partitions of three typical

porous structures, a seawall with toe protection, a rubble-mound

breakwater, and a caisson structure on a rubble foundation, are

illustrated in Figure 3.1. Theoretically, the approximation

approaches perfection when the number of the partitions approaches

infinity. In each sub-domain containing one medium there is a veloc-

ity potential which satisfies the modified Laplace equation. The

velocity potential matches both kinematic and dynamic boundary condi-

tions on the boundaries separating this rectangular sub-domain from

others and on the free surface. However, only the kinematic boundary

condition is matched on a fixed impermeable boundary. In this case,

the dynamic boundary condition provides the pressure distribution on

the boundary.

The modified Laplace equation is linear in its general form.

When linear incident waves are considered, the boundary conditions

also may be linearized. The resulting linear boundary value problem

may then be expected to have a steady state solution which is peri-

odic and has the period of the forcing function, the incident wave

Figure 3.1(a). Rectangular partitions of flow domains with inclined boundaries of aseawall with toe protection.

toIV

ra

//

/

//

\\

\\

-.

\\

\

/

/\\

\// \

// \

\

\\

...

Figure 3.1(b). Rectangular partitions of flow domains with inclined boundaries of arubble-mound breakwater.

/

,

,I IIII

IFigure 3.1(c). Rectangular partitions of flow domains with inclined boundaries of a

caisson structure on a rubble foundation.

25

field. To obtain a unique solution, the induced velocity potential

in the sub-domain with one open boundary (extending to infinity) must

also satisfy the Sommerfeld radiation condition. The radiation

condition requires that there are only out-going progressive waves at

infinity (Sommerfeld, 1949).

3.2 Seawalls with Toe Protection

As shown in Figure 3.1.(a), the original flow field with a sea-

wall has been partitioned to contain a finite number of columns.

Each column is comprised of one or more layers which contain differ-

ent media. The media in each sub-domain may be anisotropic but must

be homogeneous. The top of each column is the free surface, and the

bottom is the impermeable sea bed. Two consecutive layers are sepa-

rated by a horizontal boundary which is parallel to the x-axis.

Columns are separated by vertical boundaries which are parallel

to the z-axis. The column (or the flow field) with incident waves

has an open boundary at infinity where the Sommerfeld radiation

condition is applied to the induced waves such that there are only

out-going progressive waves generated by the interaction. The radia-

tion condition reads (Sarpkaya and Isaacson, 1981)

lim J: at Ccx

(14)

where c is the phase velocity of the progressive wave. Every layer

in the column in front of the seawall contains an impermeable verti-

cal boundary where no flow can penetrate.

26

In every sub-domain (any layer in any column), there is a veloc-

ity potential which satisfies the modified Laplace equation,

re-defined from Eq. (13) as,

a20tm tm

a20

a2

+ a 0 (15)tmx

ax2 Im z az

2

where

and

a2

tmx ftax

- iStmx

2-a

tam fRuiz

- iStmz

(16a)

(16b)

for the layer m (1 <= m <= Mt) in the column R (1 <= It). S and

f are defined by Eq. (4) and Eq. (3) in general.

The velocity potential satisfies both kinematic and dynamic

boundary conditions on the free surface and the boundary separating

two consecutive layers as shown in Figure 3.2. Here etmz stands for

the porosity in the z-direction, and ztm is the elevation of the

lower boundary of the layer m in the column t measured from the sea

bed. The coordinate of the center line in the column R is denoted by

xt, and 2Ax9 is the width of the column. However, for t = 1,

xi = x2 - Ax2, Axl = 0. Since linear waves are assumed, the free

surface boundary conditions are applied at the still water level. At

the impermeable sea bed, however, the velocity potential in the bot-

tom layer is required to satisfy only the kinematic boundary condi-

tion which means that no flow penetrates the sea bed.

ant 90E1KBC. aLiz az

182 Su a2

a

g at2 Liz az

27

z = 0

1a.tl

DBC: it g at

a2.

2 2.132

+DE: atlx 22,1 + a2 0

aztlz az2

atp 309.2KBC: e a2 2,1 - e a2tlz az .22z 2,2z az

DBC: at atattl a .11.2

z = -h+z11

a2 a2°

DE: a .9ntm + adz - obra(

2tin 3z2ax

a$2..m 2D4t(m+1)

KBC: Ea 2tmz az et(m+1)zat(m+1)z az

DBC:

KBC:

a+ a+t(mwd.)

at at

D24)2,(m+1) 2

a24(nrF1)DE: a2

2.(m+i)x 32c2+ at(m+1)z

az2

a2

a.R.(4,71)2

a+Mte

2.(M2,-1)z 2,(M2,-1)z az

etz

a VI z az

3.2.(ML1) "trilitDBC: at at

2a

ARM

DE:

a2.214Z

22z 2

t+ aRM

= 0atMtx Dx2 R az

asbai

KBC:2,

a2Mtz LHtz az

- 0

z =

z = -h+z a -1)

z = -h

x = xt Ax x foR;-"-''-'Figure 3.2. Boundary conditions on the horizontal boundaries in a

column with the free surface.

28

On a vertical permeable boundary as shown in Figure 3.3, the

velocity potential in each sub-domain also satisfies both kinematic

and dynamic boundary conditions. Here eimx is the porosity in the

x-direction. However, as at the impermeable sea bed, only the kine-

matic boundary condition is imposed on the impermeable vertical

boundary. Refer to Figure 3.4.

The capital letter Otm used in Figures 3.3 and 3.4 represents

the summation of 0Bra

for different modes of waves. The reason for

this will be discussed in Chapter 4 where the solutions are pursued.

3.3 Rubble-Mound Breakwaters

The partition of the flow field with a rubble-mound breakwater

is given as Figure 3.1(b). No impermeable vertical boundary exists

in the flow field, and therefore part of incident wave energy is

expected to be transmitted through the breakwater. In this case,

there are two sub-domains which have an open boundary at infinity.

The one on the seaward side of the breakwater contains incident

waves, the other is on the leeward side of the structure and contains

transmitted waves. The Sommerfeld radiation condition, Eq. (14), is

applied to the induced reflected and transmitted waves such that

there are only out-going progressive waves at infinity in both

domains.

All columns in the flow field have a free surface on the top and

an impermeable sea bed at the bottom. Thus the boundary conditions

on the horizontal boundaries in each column are the same as those in

the flow field with a seawall as shown in Figure 3.2.

29

z = 0

z = -h+z

ampKBC: c 8.2

LTMC fax ax

DBC:atim

at

2aG(t+1)z:

c(k+1)mxa(t+1)mx ax

aG(t+oinat

z =

2?bi,

2atJ(1+1)(0+1)

KBC: E a2.1DX tax ax

e(t+1)(m+1)a(Z+1)(m+1)x3x

DBC:

z = h+zzin

ateat

at(t4-0(u+i)

at

z h.4"." \ .".".." "O." \ '1/4 N. I

X s xR +Axz = x(t+1)(L+1)

Figure 3.3. Boundary conditions on vertical permeable boundaries.

30

z = -h+zt(m_i)

KBC: E a2

tmx tom 8x

at

z = -h+z

z = -h

yy, NN ./.0"" N %. N NAN .00. NN 0"/"."'

X = Xt+Ax

I

Figure 3.4. Boundary conditions on an impermeable vertical boundary.

31

For this case, the vertical boundaries separating adjacent

columns are all permeable. The boundary conditions on all vertical

boundaries are shown in Figure 3.3.

3.4 Caisson Structures on A Rubble Foundation

Caisson structures usually include a rubble foundation and are

protected by rubble toes as shown in Figure 1.1(c). The flow field

under consideration can be partitioned as in Figure 3.1(c). Due to

the permeability of the rubble foundation and toes, part of incident

wave energy will be transmitted under the caisson.

As shown in Figure 3.1(c), there are three different column

types. One is that bounded by a free surface, an impermeable sea

bed, and two permeable vertical boundaries. The corresponding

boundary condition can be defined by Figures 3.2 and 3.3. For the

sub-domains with an open boundary, the radiation condition is applied

at the open boundary. Another type is that bounded by a free sur-

face, an impermeable sea bed, a permeable vertical boundary, and a

vertical boundary whose upper portion is impermeable and lower por-

tion is permeable. The corresponding boundary condition is defined

by Figures 3.2, 3.3, and 3.5. The last one is the column containing

the caisson. In this column, the flow domain is bounded by two

impermeable horizontal boundaries and two permeable vertical bound-

aries. The corresponding boundary condition is defined by Figures

3.3 and 3.6.

z = h +zunr1)

KBC: e 2nitro

zifix ax

z = -h÷zIm

am

KBC: a2Wiz AM x ax

DBC:

z = -h

as

at

32

= 0

4'

AM //Jr s.Z.N / z \NS,

2 +1)2-1-1)M

2,1-1)xa( 1+1)M(

it+1 )xax

ta+1)M(1+1)

at

"e 1,. \ 4%. -", e'rx = x -f-Ax = x

2 (2+1) Ax( 2+1)

Figure 3.5. Boundary conditions on the boundaries which comprise ofboth permeable and impermeable parts.

33

'gigots 3.6. $oundayl conditions onthe

Islet nndet tkle caisson.

C4

34

4. ANALYTICAL SOLUTIONS

4.1 Introduction

The boundary value problem solution procedure begins by applying

a variable-separation solution to the modified Laplace equation in

each sub-domain. This reduces the second order partial differential

equation to two second order ordinary differential equations for two

independent variables. Two coefficients from each solution to each

equation and one common separation constant are unknown. Combining

the two solutions gives the solution to the velocity potential of the

modified Laplace equation. It is shown that one of the two unknown

coefficients in the z-dependent terms can be incorporated into those

of the x-dependent terms and the five unknowns reduce to four. Thus,

four equations from four boundary conditions (or the radiation condi-

tion) are required to determine the four unknowns.

The unknown coefficient of the z-dependent terms in the bottom

layer in any column is determined first by applying the kinematic

boundary condition (KBC) on the impermeable boundary. The two bound-

ary conditions on the upper boundary of this layer are also the

boundary conditions on the lower boundary of the layer immediately

above the lower layer. The boundary conditions provide two equa-

tions. Solving the equations determines the unknown coefficient of

the z-dependent terms in the upper layer in terms of that in the

lower layer. It is interesting to note that, for a column with

multiple layers, the boundary conditions on the upper boundary of a

layer do not yield the same eigenvalues in that layer as in a column

with one layer. Rather, they solve the unknown coefficient of the

35

z-dependent term in the overlaying layer. This leaves the eigenvalue

in the lower layer undetermined and one additional equation is

required. Continuity of the horizontal flux at the ends of the hori-

zontal boundary between two consecutive layers can provide such an

equation. The relationship between the eigenvalues in two consecu-

tive layers can then be found by combining this equation with another

equation obtained by integrating either the kinematic or dynamic

boundary condition along the horizontal boundary. As the procedure

continues, the two boundary conditions on the upper boundary (i.e.,

the free surface) of the top layer in the column yield the dispersion

equation for this column.

Infinite solutions to the dispersion equation can be found.

This gives an eigenseries representation of the velocity potential in

any sub-domain with a unique medium. For practical considerations,

only a finite number of wave modes are considered in an actual compu-

tation. The number of the wave modes is increased until the series

converges. The eigenvalues are, in general, complex numbers. They

are related to the waves which are decaying or amplifying while they

are propagating. However, only the decaying waves are considered in

the solution because the amplifying waves create energy.

Complemented by the equations relating the eigenvalues in con-

secutive layers, the dispersion equation determines all the eigen-

values in one column. The dispersion equation contains functions

which are expressed in terms of the similar functions of the second

layer, which are further expressed in terms of those in third layer,

..., etc. This nested-series relationship makes it very unattractive

36

to solve the dispersion equation by Newton's method which requires

the derivatives of the functions. The Secant method approximates the

derivatives by their difference quotients. The latter method is con-

sidered as an approximation to Newton's method and applied to solve

the dispersion equation in this study. Obtaining candidate values to

the solutions and solving the dispersion equation is, however, a

time-consuming and sometimes frustrating procedure. The procedure is

described as follows. First, the corresponding dispersion equation

in clear water is solved by Newton's method to obtain N eigen-

values. Then, applying the Secant method, the clear-water eigen-

values are given as initial guesses to solve the eigenvalues corre-

sponding to a porous structure with small increments of drag and

virtual mass coefficients and porosities. The new solutions are then

updated as new initial guesses and used to solve the corresponding

dispersion equation in the porous structure with more increments of

drag and virtual mass coefficients and porosities. The procedure

continues until the desired drag and virtual mass coefficients and

porosities are reached. Whenever the procedure fails to provide N

independent solutions, the increments are reduced by one half and the

process is repeated from the beginning.

By virtue of continuity of horizontal flux at the ends of the

boundary between two consecutive layers, the eigenfunctions (the

velocity potentials) in any column can be shown to be orthogonal over

the interval from the sea bottom to the free surface. Integrating

both the KBC and the DBC on the vertical boundaries between adjacent

columns and applying the orthogonality conditions for the eigen-

37

functions, the unknown coefficients of the x-dependent terms of each

wave mode can be solved and expressed in terms of the infinite series

of those in adjacent columns. The existence of the solution to the

boundary value problem is proved when the series converges. A

computer program has been developed to facilitate the computation.

4.2 Separable Equations of Motion

In each rectangular sub-domain, the velocity potential satisfies

the modified Laplace equation, Eq. (15), and matches proper linear

boundary conditions on the boundaries parallel to one of the axes of

the Cartesian coordinate system used in this study. The linearity of

the modified Laplace equation and the boundary conditions on such

boundaries suggests a solution of the form

2,m(x,z,t (x) Zkm

(z) e-iot

(17)

for a periodic motion. Substitute Eq. (17) into Eq. (15) to obtain

2Xkm (x)

22tm

(z)

a - + a 0kmx Xkm

(x) R.= Zw(z)

for a non-trivial solution, or

where

a

Xkm (x)2

Zkm" (z)- K

2

Xtm

(x) akmz/x Tkm 717 km

aLinz

tmz /x akmx

(18a)

(18b)

(19)

and K.trn

is the separation constant which is a complex number in gen-

eral. Equation (18b) can be rewritten as

and

and

X" (x) - K2

tmXtat

(x) = 0km

Z" (z) + Z. (z) = 0tm

a

K.

tmz/x 4a1

38

(20a)

(20b)

(1) When Ktm = 0, the solutions of Eqs. (20a) and (20b) are

Xtm(x) = A x + Btm

Ztm(z) = C z + Dtm

(21a)

(21b)

respectively.

(2) When Ktm * 0, the solutions of Eqs. (20a) and (20b) can be

written as

and

Ktm(x-xt

) -Ktm

(x-xt)

Xtm(x) = Atme + Btme

ZRm(z) = CtmcosRtm(z) + iD sinR

tm(z)

respectively, where

K2mRtm(z) = (z + h - ztm)

az/x

(22a)

(22b)

(23)

Akm' Btm' CDtm, and Ktm are the unknowns to be determined by

applying proper boundary conditions imposed on the boundaries of the

sub-domain.

39

4.3 Determining the Unknown Coefficients of the Z-Dependent Term and

the Dispersion Equation

4.3.1 Columns with a free surface

In a column with a free surface, the unknown coefficients of the

z-dependent term, Eq. (21b) or Eq. (22b), can be determined by the

boundary conditions on the horizontal boundaries as shown in Figure

3.2.

Case (1). For Kim = 0, substituting Eq. (17) and Eq. (21) into

the KBC on the lower boundary, i.e. the impermeable sea bed, of the

bottom layer Mt, it is found that

ZM = 0 (24a)

for a non-trivial solution. Matching the KBC and DBC on the horizon-

tal boundary between two consecutive layers will give

and

Ckm = 0 (24b)

DRem

X. (x) = Dt(m4.1)Xx(m+1)(x)mem

for non-trivial solutions, where 1 <= m <= (M271).

or

On the free surface, applying the KBC and DBC results in

(24c)

Dkl

0 (24d)

XIll

(x) = 0 (24e)

Either (24d) or (24e) gives a trivial solution

eim(x,z,t) = 0

for all m in any column with a free surface. This result is not

40

(25)

surprising because the solution given by Eq. (24) represents a flow

whose free surface is not oscillating.

Case (2). For Kim * 0, substituting Eq. (17) and Eq. (22) into

the KBC on the impermeable sea bed will give

DLM

0 (26a)

for a nontrivial solution. In matching the KBC and DBC on the

boundary between two consecutive layers, it is found that Cim can be

incorporated into Aim and Bim for 1 <= m <= (Mi-1). This provides the

solution

Zim(z) = cosRim(z) + iQuisinftim(z) (26b)

where

gm] 54(m+1)zaL(m+1)z

aL(m+1)x

KL(m+1)

IritanAztin

+ Qx(m+1)

im "1km

cRaz

altmz

afax

KXm

+ iQR(m+1) tan Azkm

and

(26c)

Azim = Ri(m+i)(h+zim) (26d)

In the top layer of each column, the upper boundary of this

layer is the free surface. Substituting Eqs. (26b), (22a), and (17)

into the free surface kinematic and dynamic boundary conditions

yields

where

and

02 itanAz +to -Al

g= ia

tlzatlx

Kt

1 + i(qitantzto

Azto

= R (h+zto

) = R (0)

41

(27a)

(27b)

zto (27c)

Equation (27a) is the dispersion equation of the eigenvalues in the

column.

As given by Eq. (26c), Qti is expressed in terms of (42, Qt2 in

terms of Qt3, Qtm in terms of Ot(m1.1), etc. Furthermore, Qtm

contains the unknown eigenvalues Ktm and El(m+1). Thus, Eq. (27a)

involves Mt unknowns, Ktm for 1 <= m <= Mt. Therefore, other (Mt-1)

equations are required to complement the dispersion equation to

determine the Mteigenvalues.

For a flow field with no porous media, e.g. the incident wave

field beyond the structure, the dispersion equation reduces to

a2---= IC

11tan(E

t1h)g

The solution of Eq.(28) can be shown to be either

(28)

(29a)

Or

K = +iktl tl

42

(29b)

with kn. > 0. In the first case, Eq. (28) has infinite solutions.

They are found to be

(2n-3 }1 < ktin

h < (n-1)x2

(30)

for n >= 2, where the third subscript indicates the wave mode n. The

eigenvalues correspond to the evanescent modes of waves which decay

exponentially in the x-direction away from the source causing the

waves. In the second case, Eq. (28) becomes

2a

= +kRII

tah(ktll

h) (28a)

where there is only one solution, which corresponds to progressive

waves, and tah refers to the hyperbolic tangent.

Considering all possible wave modes, Eq. (28) can be rewritten

as

22= -K

tintan(K

tinh)

where

KR 11 -11(1.11

with kkil > 0, and

Ktin = +ktin

(28b)

(30a)

(30b)

43

with (2n-3)1T/2 < kielnh < (h-1 )1r for n >= 2. Note that the choice of

the minus sign in Eq. (30a) and the plus sign in Eq. (30b) is rather

arbitrary. This is because either "+" or "-" in Eqs. (29a) and (29h)

results in the same velocity potential.

In general, the dispersion equation (Eq. (27a)) is expected to

give infinite solutions (eigenvalues). A third subscript n is

therefore added to all the functions and quantities related to the

wave of mode n. For each wave mode, there is a velocity potential

(eigenfunction) which corresponds to one eigenvalue. The velocity

potential given by Eq. (26) satisfies the modified Laplace equation,

Eq. (15), and the boundary conditions on horizontal boundaries given

in Figure 3.2. According to the principle of linear superposition,

the summation of the velocity potentials also satisfies the differen-

tial equation and the boundary conditions. This gives a general

solution of the velocity potentials in any layer in a column with the

free surface. The general solution is given as follows.

03

im(x"z t) = QRmn(x,z,t)

n=1

0Lmn "(x z t) = XLmn

(x)Z mn(z) e-iat

(x-x£)-Ximn(x-xd

XRmn(x)A

R.

e mn +Linn

e

Zi (z) = cos1/4011(z) +mn

sinAtmn

(z)

KtninRitan(z) = (z+h-z

Lm)

atmfx

(31a)

(31b)

(31c)

(31d)

(31e)

44

z&Mk

= 0 (31f)

zko

= h (31g)

r k(m+1)za2.(m+1)z

ak(m+1)x

Kt(m+Unir

itanAzkmn

+ Q1.(m+l)n

iQt(m+ntanAzzmni(qmn Ltmz

aLinz

atux

Kkmn

(31h)

QLM n(31i)

AzLoin

= R2.(m+1)n

(-h+zkm

) (31j)

where

The dispersion equation is given by

2 itanAzkon

+ Qfan

g= -ia

klzaklx

KLin [1 + iQ

kintanAz

Eon

Azion

= Rtln

(-h+zto

)

The corresponding fluid particle velocity is defined as

and

a.

utm

(x,z,t) = a2R.mx ax

2

a0tm

wkm

(x'

z'

t) = akinz az

Bernoulli's equation is given by

(32a)

(32b)

(33a)

(33b)

Bo p

-at

km+ gz = 0 (34)

45

4.3.2 A column with an impermeable upper boundary

As shown in Figure 3.6., the flow field of the column containing

the caisson is bounded by two horizontal impermeable boundaries and

two vertical permeable boundaries. The unknown coefficients of the

z-dependent term, Eq. (21b) or Eq. (22b), can be determined by apply-

ing the boundary conditions on the horizontal boundaries.

Case (1). For Kul = 0, substituting Eq. (17) and Eq. (21) into

the KBC on the lower boundary, i.e., the impermeable sea bed, of the

bottom layer, it can be found that

C 01M

t

(35)

for a non-trivial solution. Incorporating Dom into ARM and Rom ,- I

the resulting velocity potential can be shown to also satisfy the KBC

on the upper impermeable boundary. The velocity potential represents

a oscillating flow which is uniform at any vertical cross section.

Case (2). For Ktm * 0, substituting Eq. (17) and Eq. (22) into

the KBC on the impermeable sea bed gives

=LH

0

for a non-trivial solution. Z (z) can then be rewritten as5/14

t

= cosRtM

(z) + sinKtm (z)Zot

0

after incorporating %it into Atml and Buy Note that

Qtmz

= 0

(36a)

(36b)

(36c)

46

Then, substitute Eqs. (17), (22a), and (36b) into the KBC on the

upper impermeable boundary. This gives

sinRtM

(-h+zt(M -1)

) = 0 (36d)

for a non-trivial solution. The solution of Eq. (36d) is given by

RtMt

(-h+zt(Mz-1)

) = nn (36e)

Thus, from Eq. (23), the eigenvalues can be found as

atM z/xK tz

w (37)

t(Mt-1)

for n >= 1. The identity in Eq. (31f) has been applied to obtain Eq.

(37). The third subscript n in Eq. (37) indicates the wave mode.

Therefore, there are infinite velocity potentials corresponding to

the infinite eigenvalues given by Eq. (37), the dispersion equation

of the eigenvalues.

Combining cases (1) and (2), the general solution for the veloc-

ity potential in the flow domain bounded by two horizontal imperme-

able boundaries can be given as, for m representing Mt,

tm(x,z,t) = Otmm(x,z,t)

n=0

Here, for n = 0,

where

(38a)

Otm0(x,z,t) = Xtmo(x)Ztm0(z) e-iat (38b)

Xtmo(x) = Atm

ox + Btm

o(38c)

Zkmo

(z) = 1

47

(38d)

and, for n >= 1, Otm is defined by Eq. (31) but with the eigenvalues

given by Eq. (37). Note that the solution with Ktm = 0 is referred

to as the "zeroeth" mode or "0"th mode.

4.4 Determining the Relationship Between Eigenvalues in the Same

Column

In each sub-domain, the velocity potential of each wave mode

contains four unknowns after incorporating one of the coefficients of

the z-dependent term into those of the x-dependent term. They are

Atmn, Btmn, Qtmn, and Ktmn. Atmn and Bt will be determined by

applying the boundary conditions on the vertical boundaries sepa-

rating columns. Similarly, Qtmn and Ktmn are to be determined by

applying the boundary conditions on horizontal boundaries.

Consider a column with only one layer and bounded by an imperme-

able sea bed and a free surface. In this case, Qtin can be deter-

mined by the KBC on the lower boundary, the impermeable sea bed.

Km, can be solved from the boundary conditions on the upper bound-

ary, the free surface.

For a column with multiple layers, Qtmn can still be determined

by the boundary conditions on the lower boundary of the layer as

shown in the section 4.3. This procedure is the same as that in the

column with only one layer. With the exception of the top layer with

the free surface, Ktmn is not solved by applying the boundary condi-

tions on the upper boundary of the layer. The boundary conditions

nevertheless determine 0-1(m-1)n> the coefficient of the z-dependent

48

term of the layer immediately above this layer (in terms of Qkmn'

Ktmn'

and12.(m-On

). To determine Ktmn' one more equation is

required and one more condition needs to be imposed to the boundary

separating two layers. This equation can be obtained by imposing

continuity of horizontal mass flux at the ends of the horizontal

boundary between layers. Continuity is granted by considering that

the horizontal flux across a vertical boundary separating two columns

is varying continuously along the boundary. This can be seen in

Figure 3.3. This continuity condition will also play a key role when

the orthogonality of the eigenfunctions (the velocity potentials) in

each column is pursued.

Referring to Figure 3.3., continuity of horizontal flux at the

ends of the horizontal boundary at z = -h + zt(m_i) reads

CO

2

gm-uxat(m-Ox L(m-un kX' (x +Ax )

n=1

2

ckmxatmx nI1

Zmn I. tanX' (x ±Ax ) Z (-h+z

k(m-1)

OD

(39a)

after applying Eq. (31) to the condition. Take the difference of the

values at the two ends to obtain

2

egm-1)xa0m-1)x n7I [Xi(m-l)n(xi"x )

CO

X' (x -Axt(m-On t t

a= e

kmxa

n=1kmx

{[Xtmn' (xk+Ax ) - X' (x -Ax Z

Imn(-h+z

t(m-1) ))ihnn

(39b)

This equation implicitly contains Kt(m_om and Ktmm. To solve for

the relationship between them, construct another equation from the

49

original DBC (or KBC) on the horizontal boundary at z = -h + zum_i)

as follows. First, integrate the DBC along the boundary as

x +Ax 4 x+Ax., 99.ft Xi &(m-l)n]dx

.fx LI-X3011dx

L at L atx - Ax z = -h+z

L(m-1)x - Ax z=-h+z

k(m-1)

Applying Eq. (31) to Eq. (40) gives

X k(m-1)n(x +Ax ) -

X£(m- 1)n(x£ -Ax I)

or

(40)

K2

2.(m-On{X'

ft.

+Ax ) - (x -Ax )1 Z (-h+zk(m-1)

) (41a)K2Lmn k. Limn k RanIan

2

ek(m-1)xal(m-1)xn=1

r

Lx t(m-l)n(x1.+Axt) Xi(m-1)n(xk-Axit)1

,

=7

K2

ta2 t(m-1)n fx,

1.(m-Ox t(m-On=1 K2

2 kmn 1 k

- X' (x -Ax )1 Ztmn (-h+z k(m-1)I

Subtracting Eq. (41b) from Eq. (39b) gives

2co Kgm-On

{[c a2

- a2kmx kmx 2.01-1)x L(m-1)x K2

n=1kmn

I

[X' (xk

) - X' ((x+Ax -AxL)1 2 (-h+z

t(m-1))1 0

Zinn Linn

(41b)

Since the velocity potentials in two consecutive layers match the

(42)

boundary conditions on the boundary between these two layers mode by

50

mode, the eigenvalues in the layers are expected to be related to

each other mode by mode too. Therefore, Eq. (42) yields

c2

K2 =2

K2

taxatmx tan 2.(m-1)x

alt(m-1)x t(m-On

(43)

The relationship given by Eq. (43) exists for any two consecu-

tive layers. This provides (Mk-1) equations for a column with Mt

layers. These equations and the dispersion equation given by Eq.

(32) will determine the Mt eigenvalues of each mode of waves in any

column with multiple layers.

4.5 Determining the Unknown Coefficients of the X-Dependent Term

The completion of the analytical solution given by Eqs. (31) and

(38) requires the determination of the unknown coefficients of the

x-dependent term. While the unknown coefficients of the z-dependent

term are solved by matching the boundary conditions on the horizontal

boundaries at a specific z, the unknown coefficients of the

x-dependent term can be determined by matching the boundary

conditions on the vertical boundaries at a specific x.

As shown in Figure 3.1, the sub-domain containing incident waves

has an open boundary at infinity. The usual requirement or boundary

condition imposed on the flow at infinity is that the flow properties

remain finite there. However, for a flow field containing periodic

waves, this requirement is not sufficient to determine a unique

solution. This is because waves are characterized by magnitude as

well as direction of propagation.

When incident waves are intervened by structures, the induced

waves may exist at infinity with finite amplitude. Furthermore,

51

their direction of travel must be out-going, away from the struc-

ture. The combined requirements of finite amplitude and propagation

direction for the induced waves result in the so-called Sommerfeld

radiation condition as given by Eq. (14). Physically, this condition

requires that there are only out-going progressive waves at infinity.

The radiation condition is a product of steady state solutions

to boundary value problems. A steady state solution represents the

flow occurring at infinite time after initiation. An initial bound-

ary value problem is different from a boundary value problem by

specifying the boundary conditions at the time when the flow starts.

When the time goes to infinity, the solution of the initial value

problem tends toward the steady state solution. In this case, the

finite value of the induced flow at infinity is sufficient to deter-

mine a unique solution.

Since the radiation condition only allows waves to propagate in

a specific direction, it can be used as a non-reflecting boundary

condition in an initial boundary value problem solved numerically

(Orlanski, 1976). In this case, however, users must be aware of that

it is the function of the condition rather than the condition itself

that has been applied.

In the sub-domain containing incident waves, the induced veloc-

ity potential is given by Eq. (31) with m - 1. The total velocity

potential in this region can then be rewritten as

01

tZ1 "(x z t) 41(x,z,t) + / 9Lin(3(

'

z'

t)

nal

where +I represents the velocity potential of incident waves.

(44)

For n = 1, the corresponding velocity potential can be written

as

52

-i[kiii(x-xt)+at] i[ktll

(x-x2.

)-at]

SZ11(x'z't) {Arne+B

ille 1Z

I11(z)

(45)

by applying Eq. (30a) and Eq. (31c) to Eq. (31b). Substituting Eq.

(45) into Eq. (14), the radiation condition, at x + -a , it is found

that

Bill

= 0 (46)

For n >= 2, substituting Eq. (30b) into Eq. (31b), the second term of

Eq.(31c) goes to infinity as x + -a. This requires that

B tin - 0

for n >= 2.

For an incident wave given by

i(ktll

x-at)nI(x,t) = A

Ie

(47a)

(48)

where AI is the amplitude, the corresponding velocity potential$,

can be written as

-Ktll

(x-xt)

Ztll

(z) e-iatfi(x,z,t) - Btil e

by re-defining

AI 1g ik

tllxt

Btll

i ea Ch(k

tllh)

(49)

(47b)

53

Then the total velocity potential, Eq. (44), in this region can be

rewritten as Eq. (31) but with the known Bun given by Eq. (47) and

the unknown Akln to be determined by matching the boundary conditions

at x = xi + Axi, = 1. Note that xl = x2 - Ax2, Axi = 0.

4.5.1 Relationship between the unknown coefficients of the

x-dependent term in the same column

As long as the eigenvalues of the different layers in the same

column are related to each other mode by mode, the unknown coeffi-

cients of the x-dependent term in one layer can also be solved in

terms of those in any other layer in the column.

First, rewrite Eq. (41a) as

ct(m-1)xa2t(m-1)x Irt(m-1)n(xt+Axt) Xi(m-1)n(xl-Axt)1

2 r=

E£mxa£mx[Ximn(xtaxt) - X'

oi

- )1 Zt(m-1)Ln(x Ax

t tmn(-h+z ) (50)

by referring to Eq. (43). In addition, the DBCs at the ends of the

horizontal boundary at z = -h+zi(m_i) read

Xt(m -1)n

(xt±Ax ) = X

tmn(x

ttAx ) Z

tmn( -h+z

t(m -1)) (51)

From Eq. (51), it is determined that at the two sides of the column

xt(m-un

+Ax ) - Xgm-en(xt-Ax )

= [Xtmn

(xt+Ax

t) - X

Damn(x

t-Ax

t)1 Z

tmn(-h+z

t(m -1)) (52)


1

K2.(m-1)n

sh(Axt(m-1)n) lAt(m-On ER(m-1)111

sh(axImn

) [Atmn+ Bin

n]

imn(-h+z

t(m-1))K

tmn


where

sh(Axt(m-1)n)

Agm-un - B t(m-1)n]

54

(53a)

= sh(axtmn) [Atm - Bum] zain(-h+zt(m_1) ) (53b)

ax = KC

axain

Solving Eq. (53a) and Eq. (53b) results in

Atmn 2 sh(Ax

tmn) Ztm

n(-h+z

t(m-1))

1sh(axt(m-1)n)

1

( nKIran

[At(m-On kKgm-On+ 1) Et(m-l)n (K

t(m-l)n

and

sh(axt(m-1)n)1 1B =

limn 2 sh(axtmn

) Zenn(-h+z

t(m-1))

(54)

1)] (55a)

Ktmn

KInn

[At(m-1)n (K

t(m-On1) + B

t(m-1)n (Kt(m-On 1/1

(55b)

55

According to Eq. (55), the unknown coefficients of the

x-dependent term in any layer can be written in terms of those in any

other layer in the same column. For instance, it is found that,

after some tedious algebra,

sh(Ax ) K1 N,

Atmn2 sh(Axtln

(IIimn)-1 r

L tlnA (Kim 1) 4- B

Lmn111tln K

tlnbun' tin

(56a)

and

sh(Ax ) Ktmn

K.1 tin -1(Hum) (Ann (K 1) + Bun (KLmn + 1))B

turn 2 sh(Axtmn

)tin tin

where

IItln

= 1

and, for 2 <= m <= Mx,

(m-1)

IItmn

= n zgj+l)n (-h+z )

where II represents the product of the subscripted functions.

Furthermore, Eq. (50) can be rewritten as

r

ERmxatmxLXimn(xt+Axt) - Ximn(xt-Ay]

(56b)

(57a)

(57b)

= (IItmn

)-1 c a2tlx

[Xtln' (xt+Ax

t) - XtIn' (x -Ax

t)) (58)tlx

Equation (51) can also be rewritten as

n(x tAx ) = (II ) 1 X (x ±Ax )

56

(59)

4.5.2 Specific conditions for seawalls with toe protection

As shown in the previous section, the unknown coefficients Atmn

and Btmnin layer m of any column with multiple layers can be

expressed in terms of those in any other layer of the same column.

This reduces the number of the unknown coefficients of each wave mode

to only two in any column. The exception is the region containing

incident waves where one of the coefficients has been solved by Eq.

(47).

To solve the unknown coefficients in each column, the boundary

conditions on the vertical boundaries containing the column will be

applied. Referring to the eigenfunction in an arbitrary column, the

boundary conditions imposed on the boundary between this column and a

neighboring column involve the eigenfunction in the neighboring

column. Effectively, the KBC and DBC on the common boundary will

provide one equation for the eigenfunction in each column. Thus, for

any column bounded by two vertical boundaries, two equations can be

constructed from the boundary conditions on the vertical bounda

ries. For the region containing incident waves, one equation can be

obtained from the conditions on the boundary between this region and

the next column.

However, as shown in Figure 3.3 (and/or Figure 3.4), these

boundary conditions involve infinite pairs of unknown coefficients

Atmn

and BZinn

corresponding to infinite wave modes. This difficulty

57

can be resolved if the eigenfunctions in any column can be proved to

be orthogonal. The orthogonality of the eigenfunctions means that

each wave mode does not "interact" with another wave mode. The con-

dition of orthogonality will eventually provide two equations from

two vertical boundaries for the two unknown coefficients Again and

Btmn of each wave mode in each column.

The number of the equations constructed from the boundary condi-

tions is shown in Figure 4.1, where "D" represents the equation

obtained from the DBC, and "K" from the KBC.

The equation constructed from the DBC can be obtained by

performing the following steps:

Step (1) Multiply the DBC at x = xx - Axx by, for each layer,

(cla)&mu

(xt-Ax ) Z

Lad(z)

tmx

(60)

Step (2) Integrate the result of (1) from the bottom to the top of

each layer, i.e.

-10-1)INf A.111-al [(DBC at x x -Ax

t) Eq. (60)] dz

-h+ztm

Step (3) Sum the integration of Eq. (61)

Mt

[Eq. (61)]m

m=1

for all layers in the column.

(61)

(62a)

58

z

D: The equation constructed from the DBC

K: The equation constructed from the KBC

Figure 4.1. The construction of equations from boundary conditionson vertical boundaries for the case of a seawall withtoe protection.

59

Equation (62a) can be written explicitly as

co t ztmz

{ Xtmn

(xI t

) Xt (xt-Ax

) <Ztmn

(z) Ztma

(z)>1

n=1 m=1 aux

coM(1-1) et(m+1)z

= X(t_omn(xt-Axt) [(cn=1 m=1 t(m+1)x

) xt(m+1)a(xt- Ax

(t-l)mn(z)Zt(m4.1)a(z)> (1-6mm

t

)

ctm+ (Ttmx--E) xtma(xt-Axt) <Ztma(z) Z(t_1)mn(z)>1 (62b)

where 6 is the Kronecker delta, and

-h+z<Ztm

n(z) Z

abc(z)> =

a(b-1)Z n

(z) Zabc

(z) dz (63)-h+z

tra

For a = 2, b = m, c = a * n, Eq. (63) results in

<Z (z) Ztma(z)>

a

= i( 2 Ira? ) [KtmnQtmn KtmaQtma{Rmn

for m = 1, and

atmz/x 1r

L.

r

K2 -K2) 11-KtmnQtmn KthaQtmal

Inn tma

(64a)

[c2.(m-Ozat(m-1)zat(m-1)x]) Z , .Z

tmn(-h+z

t(m-1)(-h+z

itm-1))c

VimaLinz

atmx

[Kt(m-UnQt(m-On Kt(m-1)aQt(m-1)a]) (64b)

60

for 2 <= m <= Mx. For a detailed derivation of Eq. (63), refer to

Appendix A.

For n * a, substitute Eq. (64b) into the {} bracketed term on

the LHS of Eq. (62b) and expand the summation on the layers. This

results in

I e

(. 42i)t t tma

Xit

(xt-Ax .t lima 2.

) X (x -Ox ) <2 (z) 2 (z)m=1

etmx

etlz

= (rRix --) Xtin

Xtin

I(

ezzA,j(

t2x) tin 112a"

1.

2 2Klin

-Ktla

i.2 2

K12n

-Kt2a

atlz

)(atlx

)(KL1nQt1n-KtiaQtla)}

)(etlz

aR1z

2

atlx1Z

t2n(-h+z

Rd)

et2z at2x

Z1.2a

(-h+zIl)(K

LinQRan

-Kt1a

QAda

)1

ei

auz+ (411a)X. xt og

2 2 )(atmx)(Ktion(Itmn-KLmaQtmail

Rmxman m

Kt mn-Ktma

(t(m+1)z)x i

)2.(m+i)x 1.(1114)n

X t(m+1)al(K2

-K2t(m+l)n2.(m+1)a

ee

e a aZmz tmz tmx

a2

,g2.(m+l)n

(-h+ztm

)2t(m+1)a

(-h+ztm

)

et(m+1)z it(m+1)x

(Ktmn

QIan

- ELima

QRata

)1

61

et(mi -1)z+ (ke

L(ML -1)X)x

L(M1 -1)n

X

I(ML -1)a

I(

q(Mi-1)n-K:(Mt -1)aat(M-1)z

(=t(ML-1)x(MR

-1)x) -1)n-KR (M1-1)mQt(M

t-nail

ELMz

et(mit-1)z

al(MX-1)z

at(M

t-1)x

(e Li

)X2.1.1 nXIM m1( 2 2 )(Vie L E 'Kw

R

a-Ylmt

e a2 )

flitz tM

tx

ZtMe(-h+zM -1) )2.

01 a(-h+z t(Mt-1)( Et

(Mt-OnQi

(m2.-1)a

-Kt (mt-i)aSt(mcoa)}

Mt

mI=1

{ it a a axXL j tme( anStaa (240 (ks)

[( 1 )(K2 1K2 ) ( )(1

)1122 v2

ctmxatmx -Itmn-tmm st(m+1)xat(m+1)x -t(m+l)n-1(m+1)n

0 (65)

where the arguments of the x-dependent terms have been omitted. Note

that Eqs. (31i), (52), and (43) have been used in the above proce-

dure. This proves that the orthogonality of the eigenfunctions in

any column does exist in the interval from the sea bed to the free

surface. From Eq. (65), Eq. (62b) reduces to

I (i

eTax

---EXtaa(xl-AxL)] 2ataa(z)Zitma(z)

m=1 Rua

= The RHS of Eq. (62b) (66)

62

Another equation can be constructed from the KBC by performing

the following steps:

Step (1) Multiply the KBCs at (a) x = xt + Axt and (b) x = xt - Axt

by, for each layer,

co

(71;9A]cLoax

aLmx2 [XLma' (x +AxL)-XLma' (x

t-Ax )]2

Lma(z)

tmx(67)

Step (2) Integrate the results of (1) from the bottom to the top of

each layer, i.e.

-h+zof

f "`m-s' [(KBC at x = xLtAx ) Eq. (67)] dz

-h+ztm

Step (3) Sum the integration of Eq. (68)

Mk

2 [Eq. (68)]m

m=1

for all layers in the column.

Step (4) Take the difference of the results of Eq. (69) at two

boundaries, i.e.

(68)

(69)

[Eq. (69) at x = xt + Axt] - [Eq. (69) at x = xt - Axt] (70)

Equation (70) can be written explicitly as

6r XII1Z.ra a2 12r ,

RmxL Lxilan(xt+Axt)-Xi (x2.-Axt)]

mnn=1 mel Lmx

[X' (x +Ax )-X' (x -Ax )1<2tmn

(z)ZLola

(z)>IEdna tma

w MRy le

leo ',a2

nix[X' (x

t+Ax

t t)-X'

ma(x -Ax )J

&man=1 on

2Ic(X+1)(m+1)xa(t+1)(m+1)xx(t+1)(m+1)n (xl+Axt)

CZkw (z)Z(I+1)(m+l)n (z)>(1-6

)

mM(1+1)

2

c(t+l)mxa(t+l)mxX(t+l)mn(xL+Ax1)<Z(L4.1)mn(z)Zima(z)>11

=M(t-1)

2

n=1 m=1'2 c (1.-1)mxa(L-1)mx X(0.-1)mn

(x -Axt)

2{E

i(111+1)Zai(1114-1)X[Xi(M+1)a(Xt441X1)-Xion+1)3(XL-AXL)1

(1-1)mn(z)Z

t(m+1)a (z)>(1-6mM)

+ e a2

[X' (x +Ax )-X' (x )[<Z (z)Ztme tux lei t tura t tea (t-1)mb(z)>}

63

(71)

For n * a, developing a similar expansion as Eq. (65), the

bracketed term on the LHS of Eq. (71) can be shown to be zero. In

this case, Eq. (31i), Eq. (50), and Eq. (43) are applied. This

reduces Eq. (71) to

M

I

etraz

a2 Di (x +Ax )-Xi (x -Ax )]}2<Z (z)Z (z)>

M=1 XMXtmx 20X ma it .t ma t t tma

= The RHS of Eq. (71) (72)

64

The use of Eq. (50) and Eq. (43) in the procedure to obtain Eq.

(72) demonstrates the importance of requiring continuity of horizon-

tal flux at the ends of a horizontal boundary between two layers.

Substituting Eqs. (31c), (59), and (56) into Eq. (66), after

some tedious operations, it is found that

-Ax Axlia)A

ila+(e lia)B

kin La

am(E-1)

= {[ YDM(+)1A(L-1)1n

4YDM(-)]11(5-1)1n

1

1n1

nsl mal Loon lemon

where

IIIta

I [(-111a)(II )-2

<Z (z)Z (z)>]mal 11

ima Lica Zu0

YOM(t)KID

01-1)1n

-liIN CsMan]

mn t tah(Ax(L-1)mn)][II&man (11-1)mn

tah(Ax(E-1)mn

)

sh(Ax (L-1)1n)

and

L(m+1)z)<2

(L-1)mn(z)Z

k(m+1)a (a)> (,,II/(m+1)0

Swan (zL(m+1)x

Liomm j

<Zkm0

(z)2(E-1)mu

(z)>

c mx)II

Substituting Eqs. (31c), (58), and (56) into Eq. (72), it is

found that, for E * 1,

(73)

(74)

(75)

(76)

2(A/1a

+Btla

)[IIIta

sh(Axtla

)i

coMt e

=tma

)-11[YKP(-)1A(t+1)1n-EYKP(+)18 (t+1)1n1n=1 m=1 imx titan Linen

m(t-1)

- 1

1 m=1RYKM(tman +)1A

(t-01=1[n=y (a;)])3(t-1)1n1

where

(t+1)(m+l)n<2 (z)Z (z)>

)Dtla Lma (1+1)(m+1)n

tmanKP

(+) (1±Ar(t+1)(m+l)n) II

(t+1)(m+l)ntah(Ax

(t+1)(m+l)n)

(1-6mM(14.0)

(21.1)mna

(t+l)mn(2)2

Jima(z)

(1±Va+umn ))

Dtla

(51+1)mntah(Ax(w)mn)

C 14,-1)mn-alga) (1±V tman

, sh(Ax(t_i)1n)

(79)tman

(1_1)mn) It(t_omntah(Axot_umn)

65

(77)

(78)

Ktmn

Vxmn - (1(

tan

)) tah(Axim

n)

and

2

pqrepqx

apqx

KDtla

-2

e a KRix Hz KRla

(80)

(81)

Note that, for a column in front of a seawall, the first term on

the RHS of Eq. (77) disappears automatically because the normal flux

vanishes at an impermeable seawall. For the region containing

incident waves, an equation similar to Eq. (77) can be derived from

the KBC as

(Atla

-Btla

)IIIta

= f[T(41;)]A.(2.+1)1n

-[YKP(+)]8Han (t+1)1n1

n =1

66

(82)

for I = 1, where Blida is given as Eq. (47).

For t * 1, Eq. (73) and Eq. (77) provide two equations for the

two unknowns Ana and Btla

of each wave mode in each column. For

R. = 1, Eq. (82) provides one equation for the unknown Ana of each

wave mode. After these unknowns are solved, other unknowns in

different layers can be solved from Eq. (56).

In practical computations, only a finite number of wave modes

need be included. For instance, consider N wave modes. Then,

(2I -1)N equations which involve the same number of unknowns can be

constructed from Eqs. (73), (77), and (82). The unknowns can be

determined by solving the resulting matrix equation.

4.5.3 Specific conditions for rubble-mound breakwaters

As shown in Figure 3.1(b), the flow field with a rubble-mound

breakwater contains two subdomains which have an open boundary at

infinity. One is on the seaward side of the breakwater. This sub-

domain contains incident waves, and one of the unknown Btla in this

region has been solved as Eq. (47).

The other subdomain which has an open boundary at infinity is on

the leeward side of the breakwater. In this region, the eigenvalues

67

are given by Eq. (28b) and Eq. (30). For n = 1, the corresponding

velocity potential is given by Eq. (45). Substituting Eq. (45) into

Eq. (14), the radiation condition, gives

A2.11

= 0 (83a)

as x i =. For n >= 2, substituting Eq. (30b) and Eq. (31c) into Eq.

(31b), the first term of Eq. (31b) (or Eq. (31c)) becomes unbounded

as x increases. This requires that

Ailn = 0 (83b)

for n >= 2.

The construction of the equations from the boundary conditions

on the vertical boundaries is shown in Figure 4.2. For x <=0, the

equations are given by Eq. (73) from the DBC and Eq. (77) from the

KBC. For x >= 0, Eq. (73) and Eq.( 77) can still be used after

replacing k with k*, where

and

k* = It- t + 1 (84a)

x = xt (84b)

Axt* = -Ax

Note that dxIL = 0.

(84c)

68

Z



Figure 4.2. The construction of equations from boundary conditionson vertical boundaries for the case of a rubblemoundbreakwater.

69

4.5.4 Specific conditions for caisson structures on a rubble

foundation

The flow field with a caisson structure on a rubble foundation

has two regions containing an open boundary at infinity as shown in

Figure 3.1(c). This is the same as that with a rubble-mound break-

water. Thus, Eq. (47) and Eq. (83) can be applied to the correspond-

ing region, respectively.

The construction of the equations from the boundary conditions

on vertical boundaries is shown in Figure 4.3. The similarity

between Figures 4.3 and 4.2 suggests that Eq. (73), Eq. (77), and Eq.

(84) can be applied to this case, except for the column containing

the caisson and the columns adjacent to this column.

For the column containing the caisson, the equation constructed

from the DBC becomes

Y XZinn

(x Axk.

)<Zfan

(z)ZLoa

(z)>L.

n=0

= I Xf` z_1)mn(xii-AxL)<Z(& _1)mn(z)Zima(z)> (85a)

n=1

from Eq. (38), where a >= 0 and m = Mit = M(2_1). Since

atmn(z)Zuna(z)> = 0

as shown in the Appendix A, Eq. (A19), Eq. (85a) reduces to

Xtinci(xtAxI)<Zstma(z)Zula(z)

CO

= I Xft_i) (xt-Axt)<Z rmn(z)Zula(z)> (85b)

n=1 "

70



Figure 4.3. The construction of equations from boundary conditionson vertical boundaries for the case of a caissonstructure on a rubble foundation.

71

Substituting Eqs. (31c) and (56) into the RHS of Eq. (85b) gives, for

a >= 0,

Xtract(xt-Axt)<Zsma(z)Zima(z)>

= 1[YDM*(+)1A(k-eln

+[YDM*(-)18(2._01n1n=1 Lilian than

after some algebra, where

YDM*(+)

tman

(86)

K(1.-1)mn

<Z(t-1)

mn(z)Z"ma

(z)>sh(Axr

K ± tah(Ax(k-1)

)1

II"(1-1)1n)

L(1.-1)1n (t_omntah(Ax(t_omn)

For a = 0, from Eq. (38c),

Xxmo(xt-Axt) = Atmo(xt-Axt) + Bkm°

and, from Eq. (38d),

(87)

(88a)

<Z(= fa(E-1)mz/x]2.-1)

mn(z)Zkma

(z)>L K

(L-OtansinR (1.-1)mn(-11+z/(m-1) ) (88b)

For a >= 1, from Eq. (31c),

-Axkma

AxImo

Xtma

(x -Ax ) = Akma

e + B eRan

(88c)

and <Z(t_umn(z)Zima(z)> is given by Eq. (A5) in Appendix A. Note

that m = Mt = M(k-1)*

For the columns adjacent to the column containing the caisson,

the equation constructed from the KBC, i.e., Eq. (77), reduces to

2(At1a+Bi1a)[IIILash(Ax1ia)]

21.1,z

V 0)-1n=0 IMix t

{[YKP*(-)1Af1/4+1)m 1

BMR.

an (t+1)11 BM an (i+l)n

0)

m(t+1)- X {[YKM(+)1A(1_0111 -NKM(-) }B

113 (k-1)1n1n=1 m=1 than than

(89)

because the normal flux on the caisson wall vanishes. Here, for m

representing Mt = M(t+1)'

YKF*(±) = D(2.+1)mn

e

tAx(B-1-1)mn

Ida<2

Lam(Z)i

(t+1)mn(z)> (90a)

then

for n * 0; and

and

Y *(-) = D(l+l)mn a

tme(z)Z

(X+1)mn(z)>

LiuUlan

YK10*(+) = 0

km=

for n - 0, where

2

(L+1)mo c(2.+1)mxa(L+1)mxDIle 2

CR1xatlxKZia

(90b)

(90c)

(90d)

72

73

5. ANALYTICAL SOLUTION BEHAVIOR

5.1 Material Properties

The linear drag coefficient defined by Eq. (3) includes two

empirical coefficients. For a steady, non-convective flow in large

grain permeable media, they are found to be (Ward, 1964; Dinoy, 1971;

Sollitt and Cross, 1972)

vs,6lj K

Pi

and

(91a)

2

2j

Cfjej

(91b)KPJ

.)1 /2

where the subscript j stands for the x or z direction, v is the kine-

matic viscosity of the fluid, ej is the porosity of the porous struc-

ture, Kpj is the intrinsic permeability of porous media, and Cfj is a

dimensionless turbulent coefficient.

For specific porous media, the porosity, permeability, and tur-

bulent friction coefficient can be evaluated a priori from standard

tests or from empirical expressions (Dinoy, 1971; Sollitt and Cross,

1972). Although Eq. (91) is derived from steady state concepts, it

is assumed that it accounts for the damping due to the instantaneous

velocity occurring at all phases in one wave cycle. The assumptions

which limit the application of this expression are that convective

accelerations be small and that the motion be periodic with frequency

low enough to maintain the validity of the damping terms. Thus, Eq.

(91) applies when the wave length is long with respect to wave ampli-

tude and media grain size.

For homogeneous media, Eq. (3) can be rewritten as

t +T1 f ° luil dtdil

C c, V to

3 af . = 1- {

Kv + --t1I-4- }

/2K1Ito

+T

j

12dtdV

V t0

74

(92)

after substituting Eq. (91) into Eq. (3). The velocity component

shown in Eq. (92) is the real part of that defined by Eq. (10) or Eq.

(33) for a specific layer in some column.

To apply the current theory to a prototype structure, the poros-

ity, permeability, and turbulent friction coefficient should be pro-

vided as given conditions. To estimate the porosity for a prototype

porous structure may not be too difficult. However, to measure the

permeability and turbulent friction coefficient for porous media with

a grain size larger than 4 in. may be extremely difficult. Fortu-

nately, according to the results of Ward (1964), Dinoy (1971), and

Sollitt and Cross (1972), it is shown that the permeability scales

directly proportional to the square of the length ratio, and the tur-

bulent friction coefficient, and the porosity are the same in similar

materials. Therefore, for a specific material, they can be estimated

from the experimental results for smaller size media with similar

roughness and shape.

5.2 Computation Procedures

For a structure with a specific geometry, the flow domain is

first partitioned into a group of rectangular sub-domains. The par-

tition is rather arbitrary, but a finer partition should provide a

better approximation to the original structure. For a structure con-

75

taining multiple layers of different media, the width of each column

is suggested to be less than the thickness of each layer. This will

create a step shape similar to the one shown in Figure 3.1 which has

the property of Mt <= M(A +1). However, a similar shape can always be

constructed by adding more imaginary layers in each column, and the

generalized expressions of solutions can still be applied.

As shown in the flow chart given in Figure 5.1, for a given set

of wave conditions, structure geometry, and media properties, the

computation begins with the determination of the eigenvalues in each

column. The theoretical solution involves an infinite number of wave

modes as given in Chapter 4. However, only a finite number of wave

modes can be considered in real computations. The number of modal

waves included in the solution is determined such that the solution

converges to some required accuracy. To initiate the solution, a

value for the linearized drag coefficient in each subdomain is

assumed. Then the appropriate number of eigenvalues in each column

is solved numerically from the dispersion equation (combined with the

relationship between eigenvalues in different layers) by the Secant

method (Gerald and Wheatley, 1984). Rewriting the dispersion

equation into a form as

F(y) = 0 (93a)

The Secant method provides a solution from the numerical derivative

(J+1) (j)r

iF(y(j)) Fly(j-1))1

y(])I (j) (j -1)

(93b)

76

Water depth, incident wave period and height,porous structure geometry, and media properties

Eigenvalues in clear water

Eigenvalues in porous media'

lEigenvalues which are physically allowable

Coefficients of AE1n and BL

tin and 3fan

and BEmn

'Fluid particle velocities

'Linear drag coefficients

' Computed drag coefficients = previous coefficients?

no

Update linear drag coefficients

Flow properties

I End.'

yes

Figure 5.1. The flow chart of theoretical computation procedures.

77

after j iterations. The iteration scheme will stop when the differ-

ence between y(J) and y(i+1) is less than the required accuracy or

when the function value is approximately zero. The difference quo-

tient, the numerator, of Eq. (93b) approaches the first derivative of

the function as y(i) approaches y(J-1), and the Secant method reduces

to Newton's method. To solve an equation such as the dispersion

equation given by Eq. (32a), the Secant method is superior to

Newton's method. This is because the &1n term in Eq. (32a) further

involves a term given by Eq. (31h) which makes it is very undesirable

to take the derivative of the function.

As is the case for most iteration methods, the Secant method

requires good guesses to locate the desired solutions. While the

eigenvalues of evanescent waves in a flow field without porous media

can be estimated by Eq. (30), there is almost no rule to guess the

complex eigenvalues in a column which may contain several layers of

different porous media with different porosities, virtual mass coef-

ficients, and linear drag coefficients. However, since drag is known

to cause energy dissipation, the real part of a complex eigenvalue

which characterizes the attenuation behavior in Eq. (31c) is effec-

tively a consequence of the existence of linear drag coefficients in

the dispersion equation. Thus, for a column containing a specific

porous media, perspective guesses of the eigenvalues corresponding to

a given drag coefficient can be provided by starting with eigenvalues

corresponding to a zero drag coefficient. Then the dispersion equa-

tion may be resolved with drag coefficients increasing to the desired

values.

78

The eigenvalues solved from the dispersion equation may include

those corresponding to waves which are amplifying while they are

propagating. They are the eigenvalues whose real part and imaginary

part are of the same sign when velocity potentials are given by Eq.

(31b) and Eq. (31c) and the positive x direction is set to the right,

as shown in Figure 3.1. These waves can not exist in a linear wave

field with damping and therefore must be excluded from the solutions.

The eigenvalues are used to enumerate the coefficients of Ann

and Btin in the equations obtained from the boundary conditions on

vertical boundaries of each column, namely, Eqs. (73), (77), and (82)

for seawalls and rubblemound breakwaters, and Eqs. (73), (77) or

(89), and (82) for caisson structures on a rubble foundation. Note

that the eigenvalues in the layer beneath a caisson are directly

given by Eq. (37). The resulting coefficients related to the

unknowns Akin

and Btin constitute the coefficient matrix and the

coefficients related to the known B111 given by Eq. (47b) constitute

the right hand side of this matrix equation. The matrix equation is

solved by the LU decomposition method which is a modification of the

Gaussian elimination method (Gerald and Wheatley, 1984).

After Ann and Btin are found, the remaining unknowns Atmn and

Bin for m > 1 in a column with multiple layers can be solved by Eq.

(56). In addition, the fluid particle velocity and pressure at any

position can be solved by Eq. (33) and Eq. (34), respectively, where

the velocity potential is given by Eq. (31) or Eq. (38). The real

components of velocities are extracted and substituted into Eq. (92)

to calculate linear drag coefficients. If the results are different

79

from previous computed values, the above procedures are repeated.

The integrations in Eq. (92) are integrated numerically by the

trapezoidal rule. The reflection coefficient is found as the

absolute value of the ratio of A111

and B111' and, for the case with

a rubble-mound breakwater or a caisson structure on a rubble founda-

tion, the transmission coefficient is found as the absolute value of

the ratio of B 1 1 and B111. The computation is complete when the

difference of the linear drag coefficients found in two consecutive

procedures is within the required accuracy, e.g. 2%.

A Fortran program has been developed to perform the procedure

described in this section. It is listed and explained in Appendix B.

5.3 Theoretical Results

As shown in Chapter 4, the theoretical solution is a function of

porous structure geometry, porosity, incident wave conditions, as

well as internal flow characteristics such as linear drag coeffi-

cients and added mass coefficients.

The following results demonstrate the behavior of the theoreti-

cal solution for a seawall with a toe which has been partitioned, as

shown in Figure 5.2. Porous media in the toe are assumed to be homo-

geneous and isotropic. The dependence on linear drag coefficients,

porosities, virtual mass coefficients, and toe dimension is shown as

follows.

The dependence of flow properties on the linear drag coeffi-

cient, f, is illustrated in Figures 5.3 to 5.5. Three typical linear

drag coefficients are studied. They are 0.5, 1.0, and 1.5. In

Figure 5.3, the reflection coefficient is graphed as a function of

80

z

t

'77

h

"Top"

1

d

'Toe"

Figure 5.2. Definition sketch. Broken line is the original inclinedsurface.

0.99

0.98

0.97

0.96

0.95

0.94

0.93

0.92

0.91

0.9

0.89

0.88

SEAWALL WITH TOE PROTECTION

EXEZ-0.5, SXSZ-1 0

0 0.2 0.4 0.6 0.8 1 1.2

0 FX FZ -0.5 4 FXF2-1.0h / Lo

1.4 1.6

O FXFZ-1.5

1.8 2

Figure 5.3. Reflection coefficient dependence on linear drag coefficients, where d = 0.5 h.00

82

h/Lo, where h is the water depth and Lo is the deep water incident

wave length. For very short waves, e.g. h/Lo > 1.4, reflection coef

ficients approach unity because the waves are too short to feel the

presence of the toe with d a 0.5 h. Similar results occur when the

wave length tends to infinity. In this case, the dimension of the

toe is relatively small compared to the wave length and these long

waves are apparently not affected by the toe. For the waves with

h/Lo between 0.32 and 2.0, increasing linear drag tends to decrease

reflection or increase energy dissipation. However, the increase of

energy dissipation is not proportional to the change in the linear

drag coefficient. This may be explained by the observation that

increased drag results in an increased resistance to wave penetration

as well as an increase in internal wave decay. This resistance tends

to reflect waves while the waves are attenuating. When the effect of

reflection overcomes that of attenuation, increases in drag cause

more reflection than attenuation. This phenomenon occurs when the

wave length is relatively long compared to the toe dimension, e.g. as

h/Lo < 0.32.

As shown in Figure 5.3, it is interesting to note that there are

waves, with h/Lo around 0.2, which are most efficiently attenuated in

a specific porous structure. This may be attractive to ocean engi

neers who try to design the most efficient structure for attenuating

specific waves.

In Figure 5.4, the ratio of the wave length in the area of the

toe to the undisturbed wave length in clear water is plotted with

respect to the position of different columns. As waves propagate

0 998

0.995

0.994

0.992

0.99

0.988

0.986

0.984

0.982

0.98

0.978

0.976

F=-0.5(T 2)F-0.5(T-4)

`;E:AWALL WITH TOE PROTECTION

EXE7-0.5, Sx;7-1.0

0(T=.2)X .0(T4.)

Column

5

O F-1.5(7=2)F 1 5(T ..4)

Figure 5.4. Disturbed wave length dependence on linear drag coefficients, where kh = 3.1 for T = 2 sec,kh = 1.0 for T = 4 sec, and d = 0.5 h.

84

over the toe, their length tends to become shorter and shorter as

shown in the figure. For the two waves shown in the figure, the

longer wave (T = 4 sec) is not as sensitive as the shorter wave (T =

2 sec) to the change of linear drag coefficients.

The ratio of the horizontal fluid particle velocity to the deep

water celerity at the positions marked by "Toe" and "Top" in Figure

5.2 are illustrated in Figure 5.5. For very short waves, the hori-

zontal fluid particle velocities are nearly zero at both positions as

can be expected from undisturbed linear wave theory. For very long

waves, the relative velocities tend to approach the order of H/L as

estimated from shallow water wave conditions and approach zero for

small amplitude waves. In the intermediate wave range, maximum hori-

zontal velocities are observed at positions corresponding to nodes of

partial standing waves. For a perfect standing wave occurring in

front of a vertical impermeable wall, nodes are located at the posi-

tions of (2n+1)L/4, n = 0,1 2 away from the wall, where L is the

associated wave length. Thus, waves which may produce nodes at "Toe"

and "Top" can be estimated from the given distance from the wall to

each position. They are the waves corresponding to the wave numbers

of k'h = (2n+1)1T/3 for "Toe" and k'h = (2n+1)r for "Top". They cor-

respond to the local maxima of the curves in the figure. Here, k'h

is the associated disturbed local wave number. This wave number is

given as a reference scale in Figure 5.5. However, it should be

noted that the position of nodes is a function of the wave length as

well as the phase lag between incident and reflected waves.

OA

0.09

0.08

0.07

0.06

0.05

0.04

0,03

0.02

0.01


Ex EZ a5, S 0 20

FWoe)A F-0.6(To0

I 1 1 1 1 1 1 I

0.2 0.4 0.6 0.8

F- 1.O(Toe)

X F. 1 . 0(Top)

h

1 1 I 1 I 1 1

1.2 1.4 1.6

O F-1.60.00V FA6(100

1.8 2

Figure 5.5. Horizontal fluid particle velocity dependence on linear drag coefficients, where d = 0.5 h.

86

The dependence of wave properties on porosity is illustrated in

Figures 5.6, 5.7, and 5.8. Three different porosities are studied.

They are 0.25, 0.5, and 0.75. In Figure 5.6, very short or long

waves tend to be perfectly reflected for the same reasons given for

linear drag coefficients dependence in Figure 5.3. For most of

waves, e.g. h/Lo < 0.65, decreasing porosity increases reflection

coefficient as one would expect. However, for short waves such as

those with h/Lo > 0.65, this tendency does not persist for the three

indicated porosities. Evidently, more tightly packed porous struc-

tures (with small porosities) may dissipate more energy and cause

less reflection in short waves.

In Figure 5.7, the dependence of relative wave length on poros-

ity is illustrated. As one can expect, when waves propagate toward

the seawall, their wave length becomes shorter and shorter. In addi-

tion, more tightly packed porous structures tend to shorten waves

more effectively than less tightly packed structures do. For the two

waves considered, the shorter wave is less sensitive to the change in

porosity. This may be because the shorter wave (h = 10 ft, T = 2

sec, h/L = 0.5) does not feel the toe as much as the longer wave (T =

4 sec, h/L = 0.16).

In Figure 5.8, the dependence of relative horizontal fluid par-

ticle velocities on porosity is revealed. Similar behavior to that

associated with varying linear drag coefficients in Figure 5.5 is

revealed. That is, maximum horizontal velocities will occur at "Toe"

or "Top" when the wave length and the phase lag between incident and

reflected waves are optimal to produce nodes at these positions. For

1

0.99

0.98

0.97

0.96

0.95

0.94

0.93

0.92

0.91

0.9

0.89

0.88

047

0.86


FX-FZ-0.5, 4X-SZ-1.0

0.2

E -.25

0.41111

OA OA

+ E

111111 1.2

h / Lo

1.4 1.6 1.8

0 E -.75

Figure 5.6. Reflection coefficient dependence on porosities, where d = 0.5 h.

0.995

0.99

0.985

0.98

0.975

0.97

0.965

0.96

0.955

E-..25(1 -.2)A E- .25(T -4)


FX -F2-0.5, SX -SZ- 1 .0

2 3

4- E.50(T 2)X E.-.50(T4)

Column

4 5

O E-.75(T ..(2)O Em.75(Tm4)

6

Figure 5.7. Disturbed wave length dependence on porosities, where kh = 3.1 for T = 2 sec, kh = 1.0 forT - 4 sec, and d = 0.5 h.

coCO

DA

0.09

0.08

0.07

00.06

D 0.05

0.040

0.03

0.02

0.01

1:1 E .25(Toe)A E....26(7pp)


FX FZ -0.5, sx-sz-1n 18

0 0.2 0.4 0.6 0.8

E.60(Toe)X E.60(Top)

1

h / Lo

1.2 1.4 1.6

O E.75(Toe)V E .75(Top)

1.8

Figure 5.8. Horizontal fluid particle velocity dependence on porosities, where d = 0.5 h.

90

waves with h/Lo around 0.3, fluid particle velocities decrease as

porosity increases. This is because local velocities are propor-

tional to wave amplitudes and larger porosities cause more energy

dissipation for these waves as shown in Figure 5.6. The change of

the velocities due to the change of porosity is less at "Toe" than at

"Top". This result is due to the exponential decay of flow proper-

ties with respect to water depth in a wave field.

From Figure 5.9 to 5.11, the dependence of wave properties on

the virtual mass coefficient is illustrated. Since virtual mass

coefficients represent virtual forces or inertia forces, they do not

dissipate energy as does drag. Larger virtual mass coefficients

represent higher resistance forces which most likely are due to less

permeability or smaller porosities. Thus, the dependence of wave

properties on the virtual mass coefficient should be strongly related

to the dependence on porosity. More specifically, the dependence on

virtual mass coefficients and on porosities should be inversely

proportional to each other and therefore large virtual mass coeffi-

cients correspond to small porosities. This can be seen from the

similarity between Figures 5.6 and 5.9, 5.7 and 5.10, and 5.8 and

5.11. However, although porosity is purely a function of material

physical properties, virtual mass coefficients are characterized by

both material properties and flow patterns. Thus, a one-to-one

correspondence between porosity and virtual mass coefficient is not

anticipated, and this behavior is supported by the graphed results.

The dependence of wave properties on the toe geometry is illus-

trated in Figures 5.12 to 5.15. Three different geometries are


FX..F7 C1.5, EX EZ -0.5

0.99

0.98

0.97

0.96

L 0.95

0.94

0.93

0.92

0.91

0.90

111111111111110.2 0.4

0 9 -1.0

0.6 0.8 1

+ S -1.5h ft,o

1.2 1.4

0 9 -2.0

1.6 1.8 2

Figure 5.9. Reflection coefficient dependence on vitual mass coefficients, where d = 0.5 h.


0.995

0.99

0.965

0.96

0.975

0.97

0.965

0.96

0.955

51.0(.12)A S-1.0(1 -4)

3

S-1.5(1-2)S-1.5(1-4)

Caltrnn

5

O 5-2,0(1 .a.2)5-2.0(1-4)

Figure 5.10. Disturbed wave length dependence on virtual mass coefficients, where kh = 3.1 for T = 2 sec,kh = 1.0 for T = 4 sec, and d = 0.5 h.

N.)

0.1

0.09

0.08

0.07

0.06 -

0.05 -

0.04

0.03

0.02

0.01

O III! I I I I I I i I


FX -FZ -0.5. EX -EZ-0.5 22

S- 1,0(Toe)A si.o(rop)

0 0.2 0.4 0.6 0.8 1

S1.5(Toe)X S-1 .5(Top)

h / Lo

1.2 1.4 1.6 1.8

O S.2.0(toe)S-2.0(7op)

2

Figure 5.11. Horizontal fluid particle velocity dependence on virtual mass coefficients, where d = 0.5 h.

94

illustrated. They are d = 0.25 h, d = 0.5 h, and d s 0.75 h. In

Figure 5.12, the reflection coefficient is graphed as a function of

h/Lo. In this figure, it is shown that very long waves tend to be

perfectly reflected in all three cases. Also, it is found that

reflection decreases and energy dissipation increases significantly

as the toe dimensions increase.

In Figure 5.13, the dependence of relative wave length on toe

geometry is illustrated for two waves. Both wave lengths are

shortened while the waves propagate over the toe. The reduction in

wave length increases as the toe dimensions increase.

In Figure 5.14, the dependence of relative horizontal fluid par

ticle velocities at "Top" in Figure 5.2 is illustrated as a function

of toe geometry. As the distance from this position to the wall

increases, the possible wave numbers associated to nodes at this

position increases. This is also found in Figure 5.15 where the

dependence at "Toe" is illustrated. Furthermore, as the "Top" moves

close to the free surface, the amplitude of fluid particle velocities

increases correspondingly.

0.95

0.9

0.85

0.8

0.75

0.7

OAS

0.6

SEAWALL WITH TOE PROTECTIOND(-EZ-13.5, FXFZa1 .0

0 0.2 0.4 0.6 0.8

0 d..0.26h

1 1.2 1.4 1.8 1.8 2

h / Lo+ d-0.50h 0 41-0.76h

Figure 5.12. Reflection coefficient dependence on toe geometry.1/40


EXEZ0.5. SXesSZe.1.0, FkumFZ1.0

0.99 -

0.98 -

0.97 -

0.96

0.94 -

0.93

0.92

0 A-02511(7-2)A 6.012511(7-4)

3

d-0.50h(fr2)X ded0.5011(6.4)

Column

5

O d -0.75h(T -2)V d -0.76h(T -4)

Figure 5.13. Disturbed wave length dependence on toe geometry, where kh = 3.1 for T = 2 sec, kh = 1.0 forT = 4 sec.

rn

0.12

0.11

0.1

OtO

ooe

8 0.07

000

0.05

0.04

0.03

0.02

0.01

0

SEAWALL WITH TOE PROTECTIOND142-0.5, SX S2.4 .0. FX-FI 1 .0 18

0.2 0.4 0.0 MO 1 12

h / La

O d0.261N7c0 + 6.4160N7q4O 6.01751M7q4

1.4 14 113 2

Figure 5.14. Horizontal fluid particle velocity dependence on toe geometry. At "Top." 0

8

aO

MO7

OM -

MOS

0.04

0.03

0.02

MO1

0


SXSZ-1 F/CEZ..1 .0 18

0 0.2 0.4 0.6 OA 1 1.2

h Le

O c1- 0.26h(Toe) + d0.50h(Tos)O ch.0.75h(Tos)

1.4 to IA 2

Figure 5.15. Horizontal fluid particle velocity dependence on toe geometry. At "Toe."COCO

99

6. EXPERIMENTAL STUDIES: A SEAWALL WITH TOE PROTECTION

6.1 Wave Testing Facilities

Large-scale experiments were conducted at the 0.H. Hinsdale Wave

Research Laboratory (HWRL). The experiments were conducted at a

large scale to avoid possible viscous distortion occurring in low

Reynolds number models (Sollitt and DeBok, 1976).

The HWRL is a concrete channel which is 342 ft long, 12 ft wide

and 15 ft deep. Waves are generated by a hinged flap wave board,

powered by a servo-controlled oil hydraulic piston. A 112-kw elec-

tric motor drives a 76 gpm pump to generate waves up to 5 ft high.

Wave height is controlled by board stroke which is regulated by an

LVDT feedback system. Simple harmonic waves are controlled by an

electronic function generator while random waves are controlled by a

PDP 11/23 computer system (Sollitt and McDougal, 1986).

Strain gauge pressure transducers with porous stone shields,

DRUCK PDCR 10 series, are used to sense pore pressure. Electromag-

netic water current meters, Marsh-McBirney Model 115 and 35, are

applied to sense flow velocity components in two perpendicular direc-

tions in the planes normal to their major axis. Acoustic wave pro-

filers (or displacement sensors) are used to profile the wave sur-

face.

The wave testing facilities used in this study are shown in

Figure 6.1, while the detailed positions of the pressure transducers

in the toe are shown in Figure 6.2. The seawall is simulated by the

impermeable vertical wall located at the end of the channel opposite

to that of the wave generator. The vertical wall includes the origi-

1,-12: pressure transducers

13,14: flow meters

15-17: wave profilers

0.05L< AP. < 0.45L

L: incident wave length

Wave

r generator 1;171 16ir

Figure 6.1. Wave testing facility.

Figure 6.2. Location of the pressure transducers in the toe. Unit: ft.

102

nal concrete wall and an extension to the top of the channel. The

extension is constructed from 2 in. x 4 in. frame reinforced, 3/4

inch plywood.

The toe is located in front of the seawall and constructed from

the material described in section 6.3. The toe dimensions are shown

in Figures 6.2. and 6.3. The toe is five feet high at the seawall

with a bench that extends seaward a distance of five feet. A 1:2

slope connects the bench to the sea bed.

Pressure transducers in the toe are mounted on 6 in. x 16 in. x

1/4 in. aluminum plates. Each plate can accommodate two transducers

spaced at a center line distance of 12 inches. The sensor heads are

shielded by porous stones. To avoid possible phase lag caused by

presence of air in the porous stones, the stones are first boiled to

drive air out of the stones and then kept immersed in water. The

body of each transducer is protected by a 8" sleeve attached to the

plates. The conductor attached to the transducer is protected by a

plastic hose extended to outside of the toe. Refer to Figure 6.3.

6.2 Test Procedures

Pressure transducers in the toe are located at numbered posi-

tions 1 to 10 in Figure 6.1. The transducers pairs are installed by

first excavating rock material from the intended locations. The

plates with pressure transducers are set into the excavated locations

and carefully supported by well placed rocks. Wires are buried and

extended to the side wall of the channel and then collected inside

PVC pipes which carry the wires to signal conditioning instrumenta-

tion. The positions of the transducers are located by measuring the

10

In-situ

1"1

4

8"

2" 12" 2"

b) Mounting bracket dimensions

Figure 6.3. Pressure transducer mounting bracket.

104

proposed distance of the transducers from a reference point set at an

instrumentation carriage above the channel. The water level is then

raised to cover the transducers. The container which carries

immersed porous stones is placed under water and the porous stones

are removed from the container and installed on the plate. The

installation is done in water to maintain the porous stones in a

water environment after they are boiled. Rocks are then carefully

placed to support and cover the transducers. The original distribu-

tion of the porous media has been maintained by randomly placing

rocks back into the excavation.

Velocities and pressures in the water column above the toe are

profiled by a flow meter, number 14, and a pressure transducer,

number 12, in Figure 6.1. The gauges are mounted at identical eleva-

tions on a sting suspended from the instrumentation carriage on the

top of the channel walls. The shape of the sting has been stream-

lined to cause a minimum hydrodynamic disturbance. The velocities

and pressures along the surface of the toe are profiled by repeating

tests with the sting-mounted transducers at different positions along

the surface. Reference velocities and pressures are taken at the

positions close to the seawall by flow meter, number 13, and pressure

transducer, number 11, in Figure 6.1. The gauges are mounted on a

cylindrical sting suspended along the seawall. The sting is capable

of moving up and down as well as rotating. The reference readings

are used to quantify low frequency variations in the test conditions.

105

All pressure transducers mentioned above are oriented normal to

the plane of the major axis of the wave channel to avoid stagnation

pressure contamination of the dynamic signal.

The wave surfaces at different positions are profiled by acous-

tic profilers, numbers from 15, 16 and 17 in Figure 6.1. Profiler

number 15 is fixed in front of the seawall. Profiler 16 is fixed at

a position three times the water depth away from the toe of the

porous structure. The latter position is determined by requiring

that the amplitudes of the evanescent wave modes at this position are

less than 1% of those at the toe of the structure.

The wave records taken by profilers 16 and 17 are used to deter-

mine reflection coefficients by a method developed by Goda and Suzuki

(1976). The method determines reflection coefficients from the wave

records taken by two fixed wave profilers. The distance between the

profilers is suggested to be between 0.05 L and 0.45 L, where L is

the associated wave length. For a fixed wave length, the position of

the profiler 17 is fixed.

Another method is also used to resolve the incident and

reflected wave heights. Wave frequencies are adjusted to give par-

tial standing one-half wave lengths which may be integrally divided

into the channel length. This generates partial standing waves with

fixed phase and stationary positions for the nodes and antinodes.

One wave gauge is positioned over the node and another over the anti-

node. The reflected wave height is equal to one-half the difference

between the node and antinode heights. This method provides an

instantaneous estimate to the incident and reflected wave heights.

106

Note that these quantities will change with time due to multiple

reflections off the structure and wave board.

Pressure transducers, flow meters, and wave profilers are all

tested for the linearity of input and output before the tests

begin. Data are then collected by varying water depth, wave periods,

and wave heights. For each wave condition, tests are repeated to

determine the velocities and pressures at different positions along

the surface of the toe.

6.3 Material Properties

The material used in the experiments is crushed rock with a high

percentage of fractured surfaces. This material was obtained from

the Forslund Construction Co. (Albany) rock quarry in Jefferson,

Oregon. It is identified as 3 inch to 6 inch screened quarry shale.

The toe of the seawall, as shown in Figure 6.1 was first con-

structed by randomly placing rocks with a slope of approximately 1 to

2. The surface of the toe was further graded manually. A cone-

shaped hole close to the wall was excavated and the rocks in the hole

were sampled for further analysis. The size of the sample hole was

about two feet wide on top and three feet deep. The volume of the

excavation was determined by measuring the water volume required to

fill the excavation. To do this, a plastic liner was placed against

the inner periphery of the excavation and filled with water up to the

horizontal level of the local surface. The water volume was measured

with a graduated cylinder which is accurate to 1 cm3 The procedure

was repeated twice and the average value of the volume of the excava-

tion was found to be 120,330 cm3.

107

To measure the volume of the rocks in the hole, a container was

first filled with water and the rocks were carefully placed into the

container. The water replaced by the rocks was sampled and its

volume was found to be 67,280 cm3. The porosity of the porous struc

ture is then, by definition, equal to the ratio of the volume

occupied by pores to the total volume. From the measured data, it

was therefore found to be 0.441. Furthermore, the total weight of

the rocks was found to be 187.87 Kg. The specific gravity of the

material was found as 2.79 from the measured data.

Each rock removed from the hole was weighed and its major and

minor dimensions were also measured. There were 170 pieces of

rock. Corresponding to each piece of rock, an area equivalent diam

eter was defined through the relation

d -2/

where a and b are the major and minor length of the rock, respec

tively. The size distribution of the sample was then plotted as

Curve 1 in Figure 6.4. From this curve, it is found that

and

d10 6.7 cm = 2.64 in.,

d50 = 11.5 cm - 4.53 in.,

d60 12.8 cm - 5.04 in.,

Cu = d60/d 10 = 1.91 < 5.0

100

80 -

70 -

40 -

10 -

02 4 6 8 10 12 14 18

Equivalent spherical diameter (cm)

18 20

Figure 6.4. Size distribution of porous media. Curve 1 was determined from the major and minor dimensionsof individual rocks. Curve 2 was determined from the weight of individual rocks.

oco

109

The material is considered as homogeneous since Cu < 5.0.

Also, another equivalent sphere diameter was determined from the

weight of each rock through the geometric relationship

d = OAII3y

where y = 2.79 x 103 Kg/m3 and W is the weight of the rock. The

results were plotted as Curve 2 in Figure 6.4. From this curve, it

was found that

and

d10 = 6.7 cm = 2.64 in.

d50 = 10.75 cm = 4.23 in.

d60 11.60 cm = 4.57 in.

Cu = d60

/d10

= 1.73 < 5.0

Thus, the porous media are still considered as homogeneous.

In addition, an total "average" equivalent sphere diameter de

can be calculated from the equation

3nd

e Volume of rocks6 Number of rocks

It was found that

de = 9.11 cm = 3.59 in

This is about d32 found from both lines in Figure 6.4.

110

7. COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS

7.1 Introduction

Results from the experiments described in Chapter 6 are summa-

rized in this chapter. The experimental results are compared with

the theoretical results predicted by the procedures developed in

Chapter 5 for the specified test conditions.

The linear drag coefficient defined by Eq. (92) contains two

material hydraulic properties which should be determined for the test

media. They are the intrinsic permeability Kp and the turbulent

friction coefficient Cf. According to the results of Ward (1964),

Dinoy (1971), and Sollitt and Cross (1972), Kp scales directly pro-

portional to the square of the length ratio of two porous media, and

Cf is the same in similar materials. Following the scale law, Kp and

Cf for the tested media of size 3.59 in. are determined from the

results provided by Sollitt and Cross (1972). It is found that

Kp = 4.31 x 10 5ft

2and Cf = 0.3637. These media properties, the

structure dimensions described in Chapter 6, and the specified wave

conditions are input to the computer model described in Chapter 5 to

obtain the theoretical results presented in this chapter. The toe

shown in Figure 6.2 is partitioned as that shown in Figure 5.2 with

d = 5 ft.

A range of wave conditions were selected for the experiments to

span relative wave lengths from deep to shallow water, spanning

Dean's Stream Function Cases 8 through 4. Wave periods were modified

slightly from the exact stream function case to provide nearly sta-

111

tionary standing wave envelopes over the channel length. Actual wave

conditions are identified in Tables 7.1 and 7.2.

7.2 Comparison of Experimental and Theoretical Results

Figures 7.1 to 7.3 present the experimental and theoretical

reflection coefficient as a function of h/Lo for relatively constant

values of wave steepness. In Figure 7.1, five experimental data

points may be interpreted in two different ways. Reflection coeffi-

cients corresponding to relatively short waves follow the trend of

theoretical results. In contrast, the measured reflection coeffi-

cients for relatively long waves show an unreasonable amount of

energy dissipation (about 75%). This may be caused by the presence

of second order waves in the long wave envelope nodes observed during

the experiments. This would result in an over estimation of the

nodal wave height of the partial standing wave envelope, thereby

underestimating the measured reflection coefficients. In Figure 7.1

theoretical reflection coefficients are higher than measured

reflection coefficients. Similar results are found in Figure 7.2.

However, the correlation between theoretical and experimental results

is improved for the available data. In Figure 7.3 theoretical and

experimental results agree quite well where no second order waves are

observed in the wave record.

From Figure 7.4 to Figure 7.6 the reflection coefficient is

plotted as a function of wave steepness. In Figure 7.4 experimental

and theoretical results follow the trend that the reflection coeffi-

cient decreases first as wave steepness increases and then reaches a

constant value as wave steepness further increases. In this figure,

112

Table 7.1. Stream Function Cases for h 12 Feet.

CaseT

(sec)Lo

(feet) h/LoHi Range(feet)

4 10.20 533 0.0225 1.45 to 2.40

5 6.36 208 0.0578 1.09 to 2.00

6 3.93 79 0.1512 1.13 to 2.14

7 3.42 60 0.2000 1.16 to 2.17

8 2.29 27 0.4454 1.34 to 1.74

Table 7.2. Stream Function Cases for h at 10 Feet.

CaseT

(sec)Lo

(feet) h/LoHi Range

(feet)

4 11.11 632 0.0158 1.24 to 2.65

5 5.88 177 0.0565 1.07 to 2.31

6 4.54 106 0.0946 0.80 to 2.08

7 3.16 51 0.1950 0.95 to 1.69

1

OS

0.8

O .7

0.6

0.5

OA

0.2

OA


Ir1 2ft, HI/L 0.01

O

0

a

Oi i i i i i VV

0.1 012 0.14 0.16 OAS0.02 0.04 0.06 0.08

Experimenkg

h /La+ Theo not:Dal

Figure 7.1. Reflection coefficient dependence on h/Lo.

02


h -12ft, 0.015 < 1.11/1_. < 0.020

0.9 -

OA -

0.7 41

OA -

0.5

0.4 -

0.2 -

0.1 -

0

+

0

0

0

1111111110.05 0.07 0.09 011 0.13 0.15

0 EoperinuwASh /Lo

+ The:web:al

0.17


0.19

1

0.9

0.8 -

0.7 -

OA

0.5 -

0.4 -

0.3 -

0.2 -

0.1 -

SEAWALL WITH TOE PROTECTIONh -12ft, 0.04 < < 0.05

0

6

0.15

Experimental

0.25 0.35

h / Lo+ Theoretical

0.45


1

0.9 -

0.8 -

0.7 -

0.6 -

0.5 -

0.4 -

0.3 -

0.2 -

0.1 -


h-12R, T2.294sec

A 44-44 + 41- +

0

a 0 0

00

00.04 0.044

0 Experimental

0.048 0.062

HI / L+ Theoretical

0.058 0.08 0.084

Figure 7.4. Reflection coefficient dependence on wave steepness.

rn

1

0.9

0.8

02

0.8 -

0.5 -

OA

-

0.2 -

0.1 -


M.121t, T3.425sec

0

4

al

0

a

0 a

0

0.01 0.014

0 Experimental

0.018 0.022 awe 0.03

Hi / L

+ Theoretical

0.034 0.038 0.042



h-12ft, 73.937sec

OS -

0.8 -

0.7 -

0.6 -

I. OA -

0.4 -

0.3 -

0.2 -

0.t -

0.016

00

0.02

0 Experiment,'

0.024

O

+

0

0

0.026 0.032

HI / L+ Theoretical

0.036 0.04


119

theoretical results appear to be 20% higher than experimental

results. This trend is also found in Figure 7.5, however, with more

scattering of the experimental results. In Figure 7.6 experimental

and theoretical results all indicate a similar trend for wave

steepnesses less than 0.036. However, for steeper waves, experimen-

tal results show a rapid decrease in reflection coefficient while

theoretical results continue to predict the same trend observed for

lower wave steepness.

The overprediction of reflection coefficient by the theory may

partly be due to energy dissipation caused by side wall friction in

the wave channel. Furthermore, for relatively long waves, flow was

observed to penetrate through seams in the plywood freeboard exten-

sion of the reflecting seawall. This would also cause energy dissi-

pation. Finally, the net effect of reflections in a finite tank

length is to produce a nonstationary wave process. Both methods of

measuring the combined incident and reflected wave environment assume

a stationary process.

From Figures 7.7 through 7.13 dynamic pressure amplitudes at

specified positions in the toe are nondimensionalized by that

corresponding to Druk pressure transducer number 11 set at the corner

of the bench and the seawall, Figure 6.1. In Figures 7.7 to 7.13

predicted and measured nondimensional dynamic pressure amplitudes at

the locations of the numbered pressure transducers (refer to Figures

6.1 and 6.2) are listed in a). Nondimensional dynamic pressure

amplitudes at given grid points in the toe are calculated by the

theory and shown in b). And, in order to provide an improved visual

'L20

h (ft)12.0

T(sec) H(ft)6.36 1.09

Kp(ft^2) Cf4.3E-5 0.3637

Nondiro. dyn. ores.No. 1 2 3 4 5 6 7 8 9 10Pred. 0.589 0.623 0.728 0.721 0.918 0.875 0.852 0.966 0.922 0.897Mend.

a)0.575 0.607 0.731 0.738 0.903 0.891 0.88 0.968 0.946 0.936

0.951 0.969 0.984 0.995 1.003

0.845 0.879 0.911 0.928 0.942 0.953 0.96

0.823 0.855 0.883 0.9 0.914 0.924 0.9320.744 0.783 0.807 0.838 0.866 0.882 0.896 0.906 0.9130.734 0.772 0.796 0.827 0.856 0.872 0.886 0.8% 0.903

0.657 0.699 0.727 0.765 0.789 0.82 0.851 0.867 0.881 0.89 0.8980.653 0.695 0.723 0.76 0.786 0.816 0.849 0.865 0.879 0.888 0.896

0.572 0.616 0.65 0.692 0.72 0.757 0.784 0.814 0.849 0.865 0.878 0.888 0.8950.571 0.615 0.65 0.691 0.719 0.757 0.783 0.814 0.849 0.864 0.878 0.888 0.895

b)

0.95 1.0

0.7

0.6 0.6

0.5 75% 0.607

c)

Figure 7.7. Nondimensional dynamic pressure distrubtion in thetoe: a) comparison of predicted and measured values atnumbered positions, b) predicted pressure distributionat grid points, c) predicted isobars and measured values(which have three decimals).

121

hlft)12.0

Nondim.

T (see)3.93 1.13

dyn.

H(ft )

pres.

Kp(ft^2, Cf4.3E-5 0.3637

No. 1 2 3 4 5 6 7 8 9 10Pred. 0.061 0.129 0.364 0.353 0.78 0.622 0.522 0.888 0.708 0.594

Measd.

a)

0.116 0.101 0.318 0.339 0.749 0.728 0.682 0.951 0.882 0.853

91 0. 0.963 0.991 1.023

.631 0.716 0.755 0.782 0.817 0.84 0.867

I .568 0.645 0.654 0.677 0.707 0.727 0.751

0.391 0.483 0.524 0.595 0.584 0.605 0.631 0.649 0.67

.374 0.462 0.495 0.562 0.541 0.56 0.585 0.601 0.621

0.2 0.292 0.362 0.448 0.478 0.543 0.519 0.537 0.561 0.577 0.595

0.196 0.286 0.355 0.438 0.469 0.332 0.509 0.527 0.551 0.566 0.585

0.033 0.112 0.193 0.282 0.351 0.433 0.465 0.527 0.506 0.525 0.546 0.563 0.582

0.033 0.112 0.192 0.281 0.349 0.431 0.463 0.526 0.506 0.524 0.547 0.563 0.581

b)

0.6

0.9 1.0

X 0.682 0,85 3 x

c)

Figure 7.8. Nondimensional dynamic pressure distrubtion in thetoe: a) comparison of predicted and measured values atnumbered positions, b) predicted pressure distributionat grid points, c) predicted isobars and measured values(which have three decimals).

122

h (ft )

12.0

Mond IN.

Itsec)3.42

dyn.

H (ft )

1.16

pres.

(ft^2) Cf4.3E-5 0.3637

No. 1 2 3 4 5 6 7 8 9 10Pred. 0.175 0.096 0.203 0.193 0.715 0.538 0.415 0.867 0.654 0.504

Measd.

a)0.313 0.262 0.173 0.187 0.665 0.628 0.586 0.928 0.851 0.802

0.835 0.888 0.951 0.988 1.025

0.688 0.731 0.783 0.813 0.8440.512 0.63

0.448 0.552 0.575 0.611 0.655 0.68 0.705

0.224 0.342 0.402 0.495 0.493 0.524 0.562 0.584 0.605

0.211 0.323 0.371 0.457 0.441 0.468 0.502 0.521 0.541

0.033 0.106 0.201 0.308 0.352 0.433 0.411 0.436 0.468 0.486 0.504

0.032 0.103 0.195 0.299 0.34 0.419 0.396 0.421 0.451 0.469 0.486

0.215 0.116 0.031 0.101 0.192 0.293 0.335 0.412 0.391 0.415 0.445 0.462 0.479

0.214 0.116 0.031 0.1 0.191 0.292 0.333 0.41 0.389 0.414 0.443 0.46 0.478

b)

0.5

0.3 0.4

0.1 0.2 D.173

0.2 0.1 0.0

0.313g0.262

0.165

x 0.586

0.4

0.002

c)

Figure 7.9. Nondimensional dynamic pressuretoe: a) comparison of predictednumbered positions, b) predictedat grid points, c) predicted iso(which have three decimals).

distrubtion in theand measured values atpressure distributionbars and measured values

123

h (ft)

12.0

Handle.

T (sec)

2.29

dyn.

H( ft )

1.35

pres.

Kp(ft"2) Cf

4.3E-5 0.3637

No. 1 2 3 4 5 6 7 8 9 10Pred. 0.43 0.46 0.331 0.275 0.428 0.269 0.147 0.802 0.504 0.276

Measd.

a)

0.627 0.649 0.662 0.634 0.482 0.468 0.459 0.869 0.76 0.691

541 0.624 0.867 0.972 1.038

0.196 0.05 0.404 0.465 0.647 0.725 0.774

0.158 0.041 0.302 0.348 0.484 0.542 0.579

0.46 0.331 0.129 0.033 0.227 0.262 0.364 0.408 0.436

0.378 0.272 0.107 0.027 0.175 0.202 0.281 0.315 0.337

0.495 0.475 0.321 0.231 0.092 0.024 0.142 0.163 0.227 0.254 0.272

0.439 0.421 0.285 0.205 0.082 0.021 0.122 0.141 0.1% 0.219 0.234

0.404 0.467 0.408 0.392 0.265 0.191 0.076 0.019 0.113 0.13 0.18 0.202 0.216

0.395 0.456 0.398 0.382 0.258 0.186 0.074 0.019 0.11 0.127 0.176 0.197 0.211

b)

0.6 0.7 0.8 0.9 1.0

c)

Figure 7.10. Nondimensional dynamic pressure distrubtion in thetoe: a) comparison of predicted and measured values atnumbered positions, b) predicted pressure distributionat grid points, c) predicted isobars and measuredvalues (which have three decimals).

h (ft )

10.0

T (sec)

5.88 1.07

Ill ft ) Kp(ft^2) Cf4.3E-5 0.3637

Nand im. dyn. pres.

No. 1 2 3 4 5 6 7 8 9 10Pred. 0.389 0.426 0.546 0.539 0.891 0.781 0.747 0.932 0.817 0.782

Measd.

a)

0.367 0.409 0.59 0.6 0.846 0.833 0.815 0.963 0.92 0.9

981 0.965 0.973 0.993

.711 0.751 0.872 0.858 0.865 0.883

.675 0.713 0.8 0.788 0.794 0.81

0.564 0.609 0.652 0.688 0.761 0.749 0.755 0.771

0.553 0.597 0.639 0.675 0.747 0.735 0.741 0.757

0.463 0.511 0.545 0.589 0.633 0.668 0.748 0.737 0.742 0.758

0.459 0.506 0.54 0.584 0.631 0.666 0.756 0.744 0.75 0.766

0.37 0.418 0.457 0.504 0.538 0.581 0.63 0.666 0.764 0.752 0.757 0.774

0.37 0.417 0.456 0.503 0.537 0.58 0.63 0.666 0.766 0.755 0.76 0.776

b)

1.0

0.8 0.9

0.7 .846

0.60.833

0.5

0.4

0.367

)

0.409X x

0.5 90

X 0.600

0.915

124

1.037

0.923

0.847

0.805

0.79

0.792

0.8

0.808

0.811

4'0.963

X

0.920

0.900 x

orc)


125

h (ft)10.0

Nondim.

T(sec)4.54

dyn.

H(ft)0.80

ores.

Kp(ft^2) Cf4.3E-5 0.3637

No. 1 2 3 4 5 6 7 8 9 10Pred. 0.15 0.207 0.399 0.389 0.824 0.677 0.601 0.906 0.744 0.661

Measd.

a)

0.096 0.139 0.373 0.386 0.767 0.741 0.715 0.947 0.884 0.856

.933 0.937 0.962 0.991 1.039

633 0.704 0.801 0.804 0.826 0.85 0.892

0.585 0.65 0.705 0.708 0.727 0.749 0.785

0.42 0.496 0.552 0.614 0.644 0.647 0.664 0.684 0.717

.407 0.48 0.532 0.591 0.612 0.615 0.631 0.65 0.681

0.264 0.339 0.397 0.469 0.52 0.578 0.6 0.603 0.619 0.637 0.668

0.26 0.334 0.391 0.461 0.514 0.572 0.599 0.602 0.618 0.636 0.667

0.122 0.193 0.257 0.331 0.387 0.457 0.512 0.569 0.602 0.604 0.621 0.639 0.67

0.122 0.193 0.256 0.329 0.386 0.455 0.511 0.568 0.603 0.606 0.622 0.641 0.672

b)

0.2

0.6

0.5

0.3 0.4

0.096 \0.139

0.373

0.386

0.715 0.856

0.67(

c)


126

h(ft)10.0

Nondim.

T(secl H(ft)S. 16 0.95

dyn. pres.

Rp(ft'2) Cf4.3E-5 0.3637

No. 1 2 3 4 5 6 7 8 9 10Pred. 0.387 0.315 0.046 0.043 0.685 0.475 0.322 0.873 0.605 0.41

Measd.

a)

0.517 0.472 0.21 0.188 0.528 0.495 0.442 0.906 0.793 0.723

7.827 0.831 0.915 0.978 1.088

0.337 0.494 0.653 0.657 0.723 0.773 0.86

0.286 0.42 0.518 0.521 0.574 0.613 0.682

D. 05d 0.149 0.249 0.365 0.419 0.422 0.464 0.496 0.552

0.048 0.138 0.224 0.328 0.354 0.356 0.391 0.418 0.465

0.244 0.13 0.045 0.13 0.208 0.305 0.316 0.318 0.35 0.374 0.416

0.234 0.125 0.043 0.125 0.199 0.292 0.299 0.3 0.331 0.353 0.393

0.421 0.337 0.228 0.121 0.042 0.122 0.195 0.285 0.293 0.294 0.324 0.346 0.385

0.417 0.334 0.226 0.12 0.042 0.121 0.193 0.284 0.292 0.293 0.323 0.345 0.384

b)

c)


127

impression of the pressure distribution in the toe, contours of equi-

pressure or isobars are interpolated from b) and shown in c). The

measured values at the positions of transducers, marked by "x," are

given by three decimal numbers and identified in c). Note that the

isobars shown in c) of each figure were constructed by linearly

interpolating the values at adjacent grid points shown in b). How-

ever, as indicated in Chapter 4, dynamic pressure varies exponen-

tially with respect to space. Therefore, the isobars shown in c) can

only be used as a visual reference.

Seven typical wave conditions in two different water depths are

illustrated in these figures. The actual conditions are specified on

the top of each figure.

In Figure 7.7 the results for a relatively long wave, h = 12 ft,

T = 6.36 sec, are illustrated. As shown in a), experimental and

theoretical results at the instrument positions agree with each other

very well, with a relative error less than 4.2%. Theoretical and

experimental results all indicate a pressure gradient toward the sur-

face of the toe, both in the x and z directions. For this long wave

the increase of the dynamic pressure toward the seawall can be inter-

preted as a stagnation effect as flow is blocked by the seawall.

Furthermore, as shown in b) and c), dynamic pressure decays from the

toe surface as water depth increases and then reaches a constant

value as water depth further increases. In addition, a maximum ver-

tical pressure gradient occurs at the corner of the bench and the

seawall. This appears clearly in c) as the space between the isobars

increases with respect to water depth.

128

In Figure 7.8 the dynamic pressure distribution in the toe is

illustrated for a shorter wave, h = 12 ft, T = 3.93 sec. As shown in

a) of this figure, the agreement between predicted and measured

results at the points close to the toe surface is still quite good.

However, as water depth increases, theoretical results predict a

higher rate of decaying than observed in the experiments. This

results in an increase in the difference between predicted and meas-

ured results as water depth increases. Similar to that in Figure

7.7, as shown in b) and c) of the figure, a maximum vertical pressure

gradient still occurs near the corner of the bench and the seawall.

While a node may occur at the position L/4 away from the wall in

undisturbed clear water, the nearly zero dynamic pressure at the

front edge of the toe indicates a possible node there. This can be

estimated by the undisturbed wave length (L = 65 ft) for the given

conditions.

In Figures 7.9 and 7.10 the results from two shorter wave

periods are illustrated, with T = 3.42 sec and T = 2.29 sec,

respectively, where h = 12 ft. As shown in a) or c) of the two

figures, the predicted dynamic pressure decays with respect to water

depth at a rate much faster than measured values. This results in a

large difference between predicted and measured dynamic pressures in

deeper water while the agreement between the theory and the experi-

ment is still quite good at the positions close to the toe surface.

In both cases, nodes are also found around the locations L/4 away

from the seawall. This is verified by both theoretical and experi-

mental results which show a dynamic pressure gradient toward the node

129

from both sides. Furthermore, both experimental and theoretical

results indicate a maximum vertical pressure gradient around the

corner of the bench and the seawall.

From Figures 7.11 to 7.13 results are illustrated for three wave

periods in a water depth of 10 ft; they correspond to T = 5.88 sec,

4.54 sec, and 3.16 sec, respectively. Comparisons between predicted

and measured dynamic pressure amplitudes for the first two relatively

long waves are illustrated in Figures 7.11 and 7.12. Correlation is

very good, similar to the long wave results in Figure 7.7. In

addition, however, there is a relatively high pressure area found at

the corner of the sea bed and the seawall in the cases of h = 10 ft,

as shown in c) of Figures 7.11 and 7.12. This small, high pressure

area vanishes as water depth increases due to the exponential decay

of dynamic pressure with respect to water depth. For the shorter

wave, T = 3.16 sec, a node is found above the toe, as shown in Figure

7.13. The results at h = 10 ft are consistent with the results at h

= 12 ft, indicating that a maximum vertical pressure gradient is

found near the corner of the bench and the seawall. The pressure

gradient increases as wave length decreases.

Both the experimental and theoretical results indicate a pres-

sure gradient along the slope of the toe and normal to the surface of

the bench. This suggests that velocities parallel the toe slope and

penetrate the bench. Accordingly, kinematic models such as the

Morison equation will need to incorporate lift effects to quantify

destabilizing forces on the toe slope while drag and inertia effects

will be needed to quantify destabilizing forces on the bench.

130

8. CONCLUSION

8.1 Summary

This study developed a theory which provides an analytical solu-

tion to an unsteady flow field which includes a porous structure.

The flow is induced by a small amplitude wave train. The porous

structure may contain multi-layer anisotropic but homogeneous

media. Three typical porous structures are investigated. They are a

seawall with toe protection, a rubble-mound breakwater, and a caisson

on a rubble foundation. Theoretically, however, any two-dimensional

porous structure with an arbitrary geometry can be included in this

analytical procedure.

A porous structure usually contains inclined boundaries. The

flow field with inclined boundaries is first partitioned and approx-

imated by a group of rectangular, layered sub-domains. Inside each

sub-domain, resistance forces are modeled as inertia forces, skin

friction drag and form drag. Form drag is empirically known to be

proportional to the square of local fluid particle velocities and

therefore is nonlinear. The nonlinearity is resolved by defining a

linear drag under Lorentz's condition of equivalent work. The condi-

tion requires that both linear and nonlinear drag consume the same

energy in one wave period.

The periodic small motion in porous structures is then shown to

be irrotational. This insures the definition of a single-valued

velocity potential in a flow domain containing porous media. The

velocity potential in each sub-domain satisfies a partial differen-

tial equation derived from the continuity equation. This partial

131

differential equation reduces to the Laplace equation when the porous

media are isotropic.

An eigenseries representation of linear wave theory in each sub-

domain is then solved from the imposed linear boundary value problem

by the method of separation of variables. The kinematic and the

dynamic boundary conditions on the boundary between any two adjacent

sub-domains and on the free surface are matched. The kinematic

boundary condition on any fixed impermeable boundary is satisfied.

The induced velocity potential in the sub-domain with an open bound-

ary at infinity also satisfies Sommerfeld's radiation condition.

The procedure to solve the boundary value problem begins by

applying a solution whose variables are separative in the modified

Laplace equation for each sub-domain. In each layer of any column,

one of the two unknown coefficients in the z-dependent term is incor-

porated into the coefficients in the x-dependent term. The other one

is solved by applying the boundary condition(s) on the lower horizon-

tal boundary of this layer. While the boundary conditions on the

upper horizontal boundary, i.e. the free surface, of the top layer

result in the dispersion equation, those of the lower layers result

in an equation relating the eigenvalues (or the separation constants)

in two consecutive layers. This relation is solved by combining one

of the boundary conditions on the horizontal boundary between two

layers and another equation obtained from continuity of horizontal

mass flux at the ends of the boundary. For the layer beneath a cais-

son, the boundary condition on the upper boundary also results in the

dispersion equation of the eigenvalues in that layer.

132

Infinite eigenvalues are found in each sub-domain. In the sub-

domain with no porous media, the eigenvalues are real numbers. They

represent either progressive waves without damping or evanescent

modes of waves which vary exponentially in the x-direction. In a

column with porous media, the eigenvalues are, in general, the com-

plex numbers with neither real part nor imaginary part being zero.

They represent the progressive waves which are attenuating or ampli-

fying while they are propagating. Since waves can not create energy

in an energy dissipating region, only the attenuating waves are con-

sidered.

In each column, as the eigenvalues in different layers are

related to each other, the unknown coefficients of the x-dependent

terms in different layers can also be solved to be expressed in terms

of each other. This leaves only two unknowns to be further deter-

mined in each column. They are solved by applying the boundary con-

ditions on the two vertical boundaries separating (or bounding) this

column from others. Orthogonality of the eigenfunctions in each

column has been shown to exist in the interval between the imperme-

able sea bed and the free surface. The condition of orthogonality

allows for each wave mode that two unknown coefficients be determined

by the two equations. In the sub-domain with an open boundary, one

of the two unknowns of progressive waves is determined by

Sommerfeld's radiation condition. In addition, one of the two

unknowns related to each evanescent wave mode is determined by

requiring finiteness at infinity. This completes the theoretical

procedures to solve the boundary value problem.

133

8.2 Theoretical Behavior

Theoretical results presented in Chapter 5 may be summarized as

follows.

1) Reflection coefficients decrease as the linear drag

coefficient f increases for short waves. However, a

relative minimum in reflection is indicated for

intermediate values of f, with increasing reflection

for both large and small f for relatively long waves.

2) Wave length is shortened as waves propagate over

porous media. The change of wave length increases as

linear drag and added mass increase, and as porosity

decreases.

3) Reflection coefficients decrease as added mass

decreases and porosity increases for most waves.

4) Energy dissipation increases significantly as the

porous structure dimensions increase.

5) Maximum horizontal fluid particle velocities at a

specific location occur when the disturbed waves pro

duce nodes at that position. Amplitudes of the

velocities increase as added mass increases or as

porosity decreases.

8.3 Comparison with Experiments for Seawall Toes

Both theoretical and experimental results show that the reflec

tion coefficient decreases as wave steepness increases and tends

towards a constant value as wave steepness further increases. In

134

some cases, theoretical reflection coefficients are found to be 20%

higher than experimental reflection coeffiencients, even though they

agree in trend. Better agreement is found for relatively shorter

and/or steeper linear waves. The difference may be due to difficul-

ties encountered in measuring nonstationary, multiple harmonic wave

profiles observed in long wave experiments.

Comparison of nondimensional dynamic pressures from experiments

and theory indicates very good agreement for relatively long waves.

For shorter waves, agreement is obtained for the positions close to

the toe surface. Both theoretical and experimental results show that

a maximum vertical pressure gradient occurs near the corner of the

bench and the seawall. This pressure gradient increases as wave

length decreases. Furthermore, zero values of nondimensional dynamic

pressure are found under nodes in theoretical results and verified by

experimental results.

8.4 Future Investigation

The theoretical procedures developed in this study provide a

kinematic description of the flow inside and outside a porous struc-

ture. The flow is induced by a linear wave train. Therefore, the

theoretical solution is a first order approximation to a general non-

linear problem. The general nonlinear problem is a more appropriate

description of flow behavior in shallow water where porous structures

are usually constructed. Long wave experimental observations

demonstrate the existence of nonlinear second order effects. It is

then suggested that future investigations should include long wave

nonlinear effects.

135

The material hydraulic properties, the intrinsic permeability

and turbulent friction coefficient, are required to apply the the-

ory. It is suggested that future investigations should seek rational

experimental and/or theoretical procedures to quantify these pro-

perties for a range of material types and sizes.

The results of this study provide a quantitative description of

the kinematic and dynamic environment on the slope of a variety of

porous structures. The description is to be combined with a Morison

equation stability model, proposed by Chen (1987), to yield a

rational predictive model for toe armor stability.

136

9. REFERENCES

Chen, T.M. 1987. "Stability of a submerged rubble-mound toe,"Master Thesis, Oregon State University.

Dinoy, A.A. 1971. "Friction Factor and Reynolds Number Relationshipin Flow through Porous Media," ME Thesis, AIT, Bangkok,Thailand.

Gerald, C.F. and P.O. Wheatley. 1984. Applied Numerical Analysis.3rd ed. Addison-Wesley Publishing Company, Menlo Park, CA.

Coda, Y. and Y. Suzuki. 1976. "Estimation of Incident and ReflectedWaves in Random Wave Experiments," Proc. of 15th ICCE, ASCE,pp. 828-845.

Gopalakrishnan, T.C. and C.C. Tung. 1980. "Run-Up of Non-BreakingWaves -- A Finite-Element Approach," Coastal Engineering, 4, pp.3-22, Elsevier Scientific Publishing Company, Amsterdam --Printed in The Netherlands.

Hannoura, A.A. and J.A. McCorquodale. 1978. "Virtual Mass of CoarseGranular Media," J. of Waterway, Port, Coastal and Ocean Divi-sion, ASCE, Vol. 104, No. WW2, pp. 191-200.

Hannoura, A.A. and J.A. McCorquodale. 1985. "Rubble Mounds: Numer-ical Modeling of Wave Motion," J. of Waterway, Port, Coastal andOcean Engineering, ASCE, Vol. 111, No. 5, pp. 800-816.

Hedar, P.A. 1986. "Armor Layer Stability of Rubble-Mound Break-waters," J. of Waterway, Port, Coastal and Ocean Engineering,ASCE, Vol. 112, No. 3, pp. 343-350.

Kobayashi, N. and B.K. Jacobs. May 1985. "Riprap Stability UnderWave Action," J. of Waterway, Port, Coastal and Ocean Engineer-ing, ASCE, Vol. 111, No. 3, pp. 552-566.

Kobayashi, N. and B.K. Jacobs. September 1985. "Stability of ArmorUnits on Composite Slopes," J. of Waterway, Port, Coastal andOcean Engineering, ASCE, Vol. 111, No. 5, pp. 880-894.

Kobayashi, N., A.K. Otta, and I. Roy. 1987. "Wave Reflection andRun-Up on Rough Slopes," J. of Waterway, Port, Coastal and OceanEngineering, ASCE, Vol. 113, No. 3, pp. 282-298.

Liu, P.L.-F., S.B. Yoon, and R.A. Dalrymple. 1986. "Wave Reflectionfrom Energy Dissipation Region," J. WPCO, ASCE.

137

Madsen, 0.5. 1974. "Wave Transmission Through Porous Structures,"J. of the Waterways, Harbors, and Coastal Engineering Division,ASCE, Vol. 100, No. WW3, pp. 169-188.

Madsen, 0.S. and S.M. White. 1976. "Wave Transmission ThroughTrapezoidal Breakwaters," Proc. of ICCE, ASCE.

Orlanski, I. 1976. "A Simple Boundary Condition for UnboundedHyperbolic Flows," J. of Computational Physics, Vol. 21,pp. 251-269.

Sarpkaya, T. and M. Isaacson. 1981. Mechanics of Wave Forces onOffshore Structures. Van Nostrand Reinhold Co., NY.

Sollitt, C.K. and R.H. Cross. 1972. "Wave Transmission ThroughPermeable Breakwaters," 13th ICCE, pp. 1827-1846, ASCE.

Sollitt, C.K. and D.H. DeBok. 1976. "Large Scale Model Tests ofPlaced Stone Breakwaters," Proc. of ICCE, ASCE, pp. 2572-2588.

Sollitt, C.K. and W.G. McDougal. 1986. "Ocean and Coastal StructureToe Stabilization," Proposal to Sea Grant, Oregon State Univer-sity.

Sommerfeld, A. 1949. Translated by E.G. Straus. Partial Differen-tial Equations in Physics. Academic Press Inc., Publishers, NewYork, NY.

Steimer, R.B. and C.K. Sollitt. 1978. "Non-Conservative Wave Inter-action with Fixed Semi-Immersed Rectangular Structures," 16thICCE, ASCE, pp. 2209-2227.

Ward, J.C. 1964. "Turbulent Flow in Porous Media," J. of theHydraulics Divison, Proc. of the ASCE.

APPENDICES

10. APPENDICES

Appendix A

-h+a(B -1)

<2Ann

(z)ZaBy

(z)> = f Ztmn

(z)ZaBy

(z)dz-h+z

tm

138

(Al)

From Eq. (31d), after some algebra and arrangement, the integrand can

be written as

Zi (z)Z (z)

1

tmn=

2{(1+Q Q

aBy)cos[A03(+)]+i(()

2mn+C)

aBy)sin[A0(+)] 1cos[z$(+)1

where

and

+ Z 1(1-QuinQasy)cos[A0(-)] +(Qzmn-Qmsy)sin[a(-)11cos[ze(-)]

2{(1+01mnQasy)sin[A0(+)]i(Qtmn+Qaay)cos[AB(+)}sin[z8(+)]

r

l(1-4tmnQaBy)sin[A8(-)]-1(01m

n-0aBy)cos[A6(-)] }sin[z0(-)]

e( ±)

KZinn

KaBy

,tmz/x actsz/x

A6(±) = (h-ztm

) ± (h-za$

)aa

afty

atK

tmn

mz/x a$z /x

(A2)

(A3)

(A4)

where

139

Substituting Eq. (A2) into (A1) results in

<Zbun

(z)Zaey

(z)>

1= sin[(a(13-1)Ia)e(+)11(1+Qtm

nQaey

)cos[E(+)]B(+)

-1-1(Q +QcOy

)sin[g(-)11

rrza(B-1fas+ sinn Z) -z

)6( )11(1-0ban aey

)oos[E(+)1

+ i(Qum-gasy)sinR(-)11 (AS)

E(±) a Ae(±) - [h (za(13-12)+ztm)e(±)] (A6)

(1) For e(-) a 0, i.e.

Kimn Scafraz/x aaez/x

applying L'Hopital's rule to Eq. (AS) results in, after some

rearrangement,

<Zbun

(z)Zciya(z)>

1Kum

sinkza(0_1) ztm)a ][(1+Qbin

QaBy

)costiza(13-1)Y

tmz/x

+ i(Qimn+Qasy)sinAz a(8-1)y]

(A7)

1 1(1-QftnQaBy)c°s[Ae(-)]-i(Climn-QaBy)

sin[A0(-)] }(za(0-1)-zilm)

where Eq. (31j) has been used.

(2) Similarly, for e(+) = 0, i.e.

KJCmn aBy* 0

aatmz/x aBz/x

applying L'Hopital's rule to Eq. (A5) gives

<Ztan(z)Zasy(z)>

Ktmnsinkza(3_1) -zim)a zix][(1-QininQasy)cosAz a(0-1)N

im

i(QtninQaoy)sinAza(B-1)y]

2

I

t(1+0ban

QaBy

)cos[Ae(+)]+1(Qtmn +0aBy

)

sin[Ae(1-)] 1(za(s-1) -ztm)

When a = X, B = m, y = a * n, it is found that

rza(s-1)-ztm le(±) = g(±)L 2

1= [Az

2 gm-On±Az gm-1)a]

from the definition of Eq. (31j). Thus, Eq. (A5) becomes

140

(AS)

(A9)

(A10)

(All)

<2 (z)Ztma

(z)>

acos(Azn)cos(Az )a

K2 -K2ftan(Azn)(Kn-KOnQa)

n a

+ tan(Aza)(KnQnQa-Ka)+itan(Azn)tan(Aza)

a(KnQn-K

aQa

)

(KnQa-KaQn)-1(KnQn-KaQa)1+1.K2-K

2 J

n a

after some tedious algebra, where the simplifed notations

aa tmz/x

Azn

= Azt(m-On

Aza

= Azt(m-1)a

Kn

= Kbun

Ka

= Ktma

Qn Qtmn

Qa Qtma

have been used.

Rearrange Eq. (Al2) as

<2tmn

(z)Ztma

(z)>

-iacos(Azn)cos(Az )a {Kn[itan(Azn)+Qn]

2 2Kn-K

a

141

(Al2)

(Al2a)

(Al2b)

(Al2c)

(Al2d)

(Al2e)

(Al2f)

(Al2g)

where

[ l+iQatan(Aza)]-1(n[itan(Aza)+Qn][1+iQntan(Azn)]l

ia

K2-K 2 (KnQn-KaQa)Il a

is

KZ -K2

(KnQ n-Ka

Qa

)

nais

K2-K2Zimn(-h+z2(m_1)}Zum(-h+zom_1))

n a

itan(Az_)+61.. itan(Az_u)+Q_[Kn[1.14Q u( u)] K [ u)11ntan Azn a 1+1Q

atan Az

a

142

(A14)

ztnin(-h+zt(m-1)) = cos(Az2 (m_1)n)}1+iQtmntan(Azt(m_1)n)] (A15)

has been applied.

(i) For a column with the free surface, recall Eq. (31h) with

m = (m-1) and m * 1

itan(Az R.(m-1)n) +QLaincbiz

abaz

afax

Kinn

QX(m-1)n[egm-1)z agm-1)za 2.(m-1)xKL(m-1)n

1[1+4Lmn

tan(azgm-On)

(A16)

Substituting Eq. (A16) into Eq. (A14) gives, for m * 1,

<Zsuan(z)Z

&ma(z)>

(

K2

-K2 )1811mz/x(KZmnQtmn-KimaQtma)

Ruin Ina

(et(m-1)zat(m-1)zat(m-1)x)Zmn(-h+z ,t(m_1))Z itma(-h+zt.(m-1))

2etmz

akmx

t(m-1)nQt(m-1)n-K gm-1)aQt(m-1)a)1

143

(A17a)

For m = 1, applying the dispersion equation, Eq. (32a), to Eq. (A14)

results in

ia, /<Z

gmn(z)Z

tma(z)> ( maLzIN )(K Q -K Q ) (A17b)

2 2 bin tmn Ema tmaKtm

n-Ktma

(ii) For a column with no free surface, apply Eq. (31j) and Eq.

(36e), it is found that

Az2.(m-1)n

= n7 ; Azt(m-1)a

= aw

and

(A18a)

tanAzt(m-1)n = 0 = tanAz

t(m-1)a(A18b)

Thus, substituting Eqs. (A18) and (31i) into Eq. (A14) gives

<Zimn(z)Zula(z)> = 0 (A19a)

for n * a, n * 0, a * 0. When n * 0, a = 0, it is found that

<Ztmn

(z)Zitima(z)> =

-h+zgm-1)

Ztmn

(z)dz = 0-h+Z

(A19b)

by combining Eqs. (38d), (31d), (31i), and (37). When n = a = 0,

from Eq. (38d), it is found that

<21.1110(z)2zino(n)> = zz(n_i)

where in represents Hz.

144

(A20)

145

Appendix B

$DEBUG%LARGE******************************************************** PROGRAM SW2.FOR* This program solves the potential flow in a flow* field with the appearance of a porous structure. The** structure may contain multiple layers of homogeneous** but anisotropic media.* This is designed for a seawall with toe protection. ** It is required that IL )= 3* Here, IL is the number of the column.* NOTATIONS:* WD: water depth* T: wave period* WH: wave heigh* DFZ: the suggested increment of FZ used to solve

K(L,M,N) (see: subroutine EIGENDO)* SMALLF: output criterion for FX and FZ* MML(L): number of given grid points in each column ** NNLM(L,M): number of given grid points in the layer *

M in column L.* REMARKS:

The second subscript m in the theory has been *shifted by 1. For example, 12(L,M,N) in theory *is renamed as Q(L,M +1,N) in the program and *all subroutines related.

*******************************************************COMPLEX*16 A(6,5,8),B(6,5,8),AB(88,88),CC(88)

,CI,K(6,5,8),O(6,5,8),KWD12(8),AXL(5),AZL(5),QL(5,8),KWDL2(8),AZ(6,5)

& ,II(6,5,8),CDSINH,CCA(8),CCB(8),DET,XAB(88),AX(6,5),DXLA,SHII,KM2

REAL*8 X(6),DX(6),Z(6,5),EX(6,5),EZ(6,5),SX(6,5),SZ(6,5),FX(6,5),FZ(8,5),CFX(6,5),OFZ(6,5)

& ,KPX(6,5),KPZ(6,5),KH1,KHN,ZL(5),FXL(5),SZL(5),EXL(5),EZL(5),FXX(6,5),FZZ(6,5),UA2(5,10),WA2(5,10),UA3(5,10),WA3(5,10),UA1(5,10),WA1(5,10),SIU2,SIU3,SIW2,SIW3

& ,PI,G,WD,T,WH,DFXZ,SMALLF,TIP2,GH,DFZ,HX,XVX,ZVHZ,U1,U2,W1,WE,HLO,CO,U0,FZL(5),SXL(5),NU,HZ

INTEGER ALPHADIMENSION ML(6),ML1(6),ML2(6),NML(6),MML(6)

,NNLM(6,5),NNN(6)COMMON / DAT1 / X

146

COMMON / DAT2 / DX,ZCOMMON / DAT3 / EX,EZCOMMON / DAT4 / AX,AZCOMMON / DAT5 / K,12

COMMON / DATE, / ABCOMMON / DAT7 / A,BCOMMON / DAT10 / MML,NNLMCOMMON / DAT11 / UA1,WA1,UA2,WA2,UA3,WA3COMMON / DAT12 / FXL,FZL,SXL,SZL,EXL,EZL

,ZL,AXL,AZL,OLCOMMON / DAT22 / IICOMMON / DAT61 / CC,XRBOPEN(1,FILE=0INPUT.SW2',STATUS=tOLD')OPEN(2,FILE='OUTPUT.SW,,STATUS='NEW,)

** Units: (Length:Foot), (Mass:Slug), (Time:Sec.)PI=4.0*ATAN(1.0)G=32.2CI=CMPLX(0.0,1.0)NU =1. 09E -5

AX(1,2)=CIAZ(1,2)=CI

C)))) Reading in data from "INPUT.SW":CCC READ IN DATA FROM (INPUT.SW)

READ(1,*)WDREAD(1,*)ILREAD(1,*)(ML(L),L=1,IL)DO 1,L=1,ILML1(L)=ML(L)+1

1 CONTINUEREAD(1,*)(X(L),L=1,IL)READ(1,*)(DX(L),L=1,IL)DO 201,L=2,ILMLL=ML(L)

201 READ(1,*)(Z(L,M),M=2,MLL)DO 2,L=1,ILZ(L,1)=WDML1L=ML1(L)Z(L,ML1L)=0.0

2 CONTINUEDO 202,L=1,ILML1L=ML1(L)

202 READ(1,*)(EX(L,M),M=2,ML1L)DO 203,L=1,ILML1L=ML1(L)

203 READ(1,*)(EZ(L,M),M=2,ML1L)DO 204,L=1,ILML1L=ML1(L)

204 READ(1,*)(SX(L,M),M=2,ML1L)DO 205,L=1,ILML1L=ML1(L)

205 READ(1,*)(SZ(L,M),M=2,ML1L)

147

DO 206,L=1,ILML1L=ML1(L)

206 READ(1,*) (FX(L,M),M=2,ML1L)DO 207,L =1, ILML1L=ML1(L)

207 READ(1,*)(FZ(L,M),M=2,ML1L)DO 208,L=1,ILML1L=ML1(L)

208 READ(1,*)(CFX(L,M),M=2,ML1L)DO 209,L=1,ILML1L=ML1(L)

209 READ(1,*)(CFZ(L,M),M=2,ML1L)DO 210,L =1, ILML1L=ML1(L)

210 READ(1,*)(KPX(L,M),M=2,ML1L)DO 211,L=1,ILML1L=ML1(L)

211 READ(1,*)(KPZ(L,M),M=2,ML1L)READ(1,*)NNREAD(1,*)DFXZ,SMALLFREAD(1,*)(MML(L),L=1,IL)DO 212,L=1,ILML1L=ML1(L)

212 READ(1,*)(NNLM(L,M),M=2,ML1L)CM(213 READ(1,*)T,WH

IF(T.EQ.0.0) GO TO 118TIP2=2.0*PI/TGH=TIP2**2*WD/G

C)))) Finding the eigenvalues in the clear water: L=1CCC EIGENVALUES IN CLEAR WATER: K(L,1,N),N=1,NN

CALL KHJ1(GH,KH1)K(1,2,1)=-1.0*CI*KH1/WD0(1,2, 1) =0.0DO 3,N=2,NNCALL KHJJ(N,SH,KHN)K(1,2,N)=KHN/WD0(1,2,N)=0.0

3 CONTINUE1234 FoRMAT(Ix,7*********************************1)

C((((DXLA=-1.0*K(1,2,1)*X(1)8(1,2,1)=0.5*CI*WH*G/TIP2/COSH(KH1)*EXP(DXLA)

C III=1C>>>> Finding the eigenvalues in porous media: L)=2CM Assigning the eigenvalues in clear waterC as initial guesses:

DO 4,N=1,NNKWD12(N)=K(1,2,N)*WD

4 CONTINUEC(((

148

CCC EIGENVALUES IN POROUS MEDIA: K(L,M,N),N=1,NN117 DO 5,L=2,IL

MLL=ML(L)ML1L=ML1(L)DO 6,M=1,ML1LZL(M)=Z(L,M)

6 CONTINUEDO 7,M=2,ML1LFXL(M)=FX(L,M)FZL(M)=FZ(L,M)SXL(M)=SX(L,M)SZL(M)=SZ(L,M)EZL(M)=EZ(L,M)EXL(M)=EX(L,M)

7 CONTINUEDFZ=DFXZCALL EIGENDO(MLL,NN,WD,GHIDFZ,KWD12,KWDL2)DO 8,M=2,ML1LAZ(L,M)=AZL(M)AX(L,M)=AXL(M)

8 CONTINUEDO 9,N=1,NNK(L,2,N)=KWDL2(N)/WD0(..,2,N)=OL(2,N)IF(ML1L.EQ.2) GO TO 9DO 10,M=3,ML1LK(L,M,N)=K(L,21N)*SORT(EXCL,2)/EXCL,M))

*(AX(L,2)/AX(L,M))O(L,M,N)=OL(M,N)

10 CONTINUE9 CONTINUE, CONTINUE

C((((CM> CHOOSE ONLY THE WAVES WHICH ARE PHYSICALLYC ALLOWABLE:

CALL FIL(IL,ML1,NN,NNN)CM(C)))> Calculating the coefficient matrix AB(I,J):

NML(1)=1NML(2)=NML(1)+NNN(1)DO 11,L=3,ILNML(L)=NML(L-1)+2*NNN(L-1)

11 CONTINUEMIJ=NML(IL)-1+2*NNN(IL)CALL IILMN(IL,ML1,NNN)CALL ABIJ(WD,IL,NNN,ML1,NML,MIJ,CCA,CCB)

CM(CM) Calculating CC(I), I=1, MIJ:

CC(1)= B(1,2, 1)C))) Assigning CC(I)=0.0 for all I )= 2:

DO 12,1=2,MIJ

149

CC <1)=0.012 CONTINUE

C <

CF ) ) Replacing CC(I) I ) by the correct values:DO 13, ALPHA=1, NNN (2)IA= (NML (2) -1 ) +2*ALPHA-1IB=IA+1CC ( I R) =B ( 1,2,1 )4eCCA (ALPHA)

CC ( I B) =E1( 1,2,1 )*CCB (ALPHA)13 CONTINUE

C < <

C<<(<C) ) ) ) Solving A (L, M, N) and B (L, M, N)CCC SOLVING R (L, M, N) , B (L, M, N) :C> > Solve A (L, 2,N) and B(L,2,N)

CALL MATRI XC (MI3, DET)IF (ABS (DET) . EQ. O. 0) GO TO 999L=1

1)=XAB(1)DO 14, N=2, NNN(L)R (L, 2, N) =XAB (N )

B (L, 2, N)=0.014 CONTINUE

DO 15,L =2, ILDO 16, N=1, NNN(L)IA= (NML (L) -1 ) +2*N-1IB=IA+1(L, 2, N)=XAB ( IA)

B (L, 2, N)=XAB ( IS)

16 CONTINUE15 CONTINUE

C ((<

C))> SOLVING A (L, M, N) and B (L, M, N)C FROM A (L, 2, N) AND B (L, 2,N)

DO 17,L =2, ILML1L=ML1 (L)IF (ML1L. EQ.2) GO TO 17DO 18, M=3, ML1LDO 19, N=1, NNN (L)SHI I=0. 5*CDSINH (1_, 2, N)*DX (L) ) /CDSINH

(K (L, M, N)*DX (L) ) /II (L, M, N)

KM2=K (L, M, N) /K (L, 2, N)

(L, M, N)=SHI I* (A (L, 2, N)*(KM2+1.0)a +8 (L, 2, N)*(KM2-1. 0) )

El <1_, rl, N)=SHII*CA (L, 2, N)*(KM2-1. 0)+B (L, 2,N)* (KM2+1. 0) )

19 CONTINUE18 CONTINUE17 CONTINUE

C < <

C < < <

150

CM> Calculating FX(L,M) and FZ(L,M)CCC CALCULATING FX(L,M), FZ(L,M):

DO 20, L =2, ILMMLL=MML(L)HX=2.0*DX(L)/MMLLDO 21,M=2,ML1(L)NNLMLM=NNLM(L,M)M1=M-1HZ=(Z(L,M1)-Z(L,M))/NNLMLMIF(EX(L,M).LT.1.0) GO TO 101IF(EZ(L,M).E0.1.0) GO TO 102

101 CALL UW(WD,L,M,NNN)CALL SITR(HX,HZ,MMLL,NNLMLM,UA2,SIU2)CALL SITR(HX,HZ,MMLLINNLMLM,UA3,SIU3)CALL SITR(HX,HZ,MMLLINNLMLM,WA2,SIW2)CALL SITR(HX,HZ,MMLLINNLMLM,WA3,SIW3)

777 FXX(L,M)=1.0/TIA2*EXCL,M)*(NU/KAX(L,M)+8.0/3.0/PI*CFX(L,M)*EX(L,M)/SURT(KAX(L,M))*SIU3/SIU2)

FZZ(L,M)=1.0/TIP2*EZ(L,M)*(NU/KPZ(L,M)+8.0/3.0/PI*CFZ(L,M)*EZ(L,M)/SCART(KAZ(L,M))*SIW3/SIW2)

GO TO 21102 FXX(L,M)=0.0

FZZ(L,M)=0.021 CONTINUE20 CONTINUE

C((((C)))) Testing if the FX, FZ are the same in twoC consecutive computations:

L=2115 M=2113 TEST=ABS(FX(L,M)-FXX(L,M))/FX(L,M)

IF(TEST.GT.SMALLF) GO TO 111TEST=ABS(FZ(L,M)-FZX(L,M))/FZ(L,M)IF(TEST.GT.SMALLF) GO TO 111IF(M.EO.ML1(L)) GO TO 112M=M+1GO TO 113

112 IF(L.EO.IL) GO TO 114L=L+1GO TO 115

C))) If not, update FX, and FZ111 IF(III.GE.50) GO TO 116

III=III+1DO 22,L=2,ILDO 23,M=2,ML1(L)FX(L,M)=FXX(L,M)FZ(L,M)=FZZ(L,M)


151

GO TO 117C<(<

116 WRITE(*,'(A)1)1 # of iterations is over 50'C)>) If yes, write solutions into "OUTPUT.SW":

114 WRITE(,*)(X(L),L=1,IL)DO 501,L=1,ILML1L=ML1(L)WRITE(2,*)(Z(L,M),M=E,ML1L)WRITE(2,*)(FX(L,M),M=2,ML1L)WRITE(E,*)(FZ(L,M),M=E,ML1L)WRITE(2,*)(AX(L,M),M=2,ML1L)

501 WRITE(E,*)(AZ(L,M),M=2,ML1L)DO 502,L=1,ILNNNL=NNN(L)DO 502,M=2"1(L)WRITE(2,*)(A(L,M,N),N=1,NNNL)WRITE(2,*)(B(L,M,N),N=1,NNNL)WRITE(2,*)(K(L,M,N),N=1,NNNL)

502 WRITE(2,*)(O(L,M,N),N=1,NNNL)WRITE(2,*)WDWRITE(,*)IIIGO TO 118

C<<<C(<<(

999 WRITE(*,,(A)9)' THE MATRIX IS SINGULAR !!!'118 CLOSE(1)

CLOSE(2)END

*****************************************************SUBROUTINE ABIJ(...)

* This subroutine calculates the elements of the* coefficient matrix of A(L,2,N) and B(L,2,N).* This is designed for SW2.FOR.*****************************************************

SUBROUTINE ABIJ(WD,IL,NN,ML1,NML,MIJ,CCD,CCK)COMPLEX*16 AB(88,88),K(6,5,8),O(6,5,8),AX(6,5)

,CCD(8),CCK(8),EAK(6,5,8),VRT(6,5,8),IIIK(6,8),IIID(6,8),IIIKL,IIIDL,ABKA,ABDA,ABDB,YKAA,YKAM,YKMA,YKMM,YDMA,AZ(6,5),III(6,5,8),ABKB,YDMM,EZXII

RERL *8 EX(6,5),DX(6),Z(6,5),EZ(6,5)INTEGER ALPHA,AJA,BJA,AJM,BJMDIMENSION ML1(6),NML(6),NN(6)COMMON / DATE / DX,ZCOMMON / DAT3 / EX,EZCOMMON / DAT4 / SX,AZCOMMON / DATS / K,OCOMMON / DATE, / ABCOMMON / DAT21 / EAK,VKTCOMMON / DAT22 / IICOMMON / DATE3 / IIIK,IIID

152

CALL EAKVKT(IL,ML1,NN)CALL IIIKD(IL,ML1,NN)

C)))) COEFS OF A(L,2,ALPHA) AND B(L,2,ALPHA)C))) COEFS OF A(ID,2,ID),A(IK,21IK), AND B(IK,2,IK)

DO 1,I=1MIJDO 2,J=1,MIJIF(I.EQ.J) GO TO 100AB(I,J)=0.0GO TO 2

100 AD(I,J)=1.02 CONTINUE1 CONTINUE

CMC))) COEFS OF A(IK,2,IK-1) AND B(ID,2,ID+1)

DO 3,L=2,ILDO 4,ALPHA=1,NN(L)ID=(NML(L)-1)+2*ALPHP-1IK=ID+1IDP1=ID+1IKM1=IK-1AB(IK,IKM1)=1.0AB(ID,IDP1)=EXP(2.0*K(L,2,ALPHA)*DX(L))

4 CONTINUE3 CONTINUECM

CM(C)))) COEFS OF AB(I,J), WHATEVER ELSEC))) FOR L=1, COEFS OF A(L+1,2,N) AND B(L+1,2,N)

L=1M =2

DO 5,ALPHA=1,NN(L)IK=ALPHADO 6,N=1,NN(L+1)AJP=(NML(L+1)-1)+2*N-1DJP=AJP+1CALL YKAAM(WD,L,M,N,ALPHA,ML1,YKPP,YKPM)AB(IK,AJP)=-1.0/IIIK(L,ALPHA)*YRAMAB(IK,DJP)=1.0/IIIK(L,ALPHP)*YKPA

B CONTINUE5 CONTINUE

CMC)>) FOR L > =2

DO 7,L=2,ILML1L=ML1(L)ML1LM=ML1(L-1)DO 6,ALPHA=1,NN(L)ID=(NML(L)-1)+2*ALPHA-1IK=ID+1IIIDL=IIID(L,ALPHA)IIIKL=IIIK(L,ALPHA)

C)) COEFS OF A(L-1,2,N) AND B(L-1,2,N)

153

DO 9,N=1,NN(L-1)IF(L.E0.2) GO TO 101RJM=(NML(L-1)-1)+2*N-1BJM=AJM+1GO TO 102

101 RJM =N102 ABKA=0.0

ABKB=0.0ABDA=0.0ABDB=0.0DO 10,M=2,ML1LMCALL YKDMPM(WD,L,M,N,ALPHA,ML1,YKMP,YKMM

,YDMP,YDMM)ABKA=ABKA+YKMPABKB=ABKB+YKMMABDA=ABDA+YDMAABDB=ABDB+YDMM

10 CONTINUEIF(L.EQ.2) GO TO 103AB(ID,AJM)=-1.0/IIIDL*ABDAAB(ID,BJM)=-1.0/IIIDL*ABDBAB(IK,AJM)=0.5/IIIKL*ABKAAB(IK,BJM)=-0.5/IIIKL*ABKBGO TO 9

103 AB(ID,AJM)=-1.0/IIIDL*ABDAAB(IK,AJM)=0.5/IIIKL*ABKAIF(N.GT.1) GO TO 9CCD(ALPHA)=1.0/IIIDL*ABDBCCK(ALPHA)=0.5/IIIKL*ABKB

9 CONTINUEC<<C)> COEFS OF A(L+1,2,N) AND B(L+1,2,N)

IF(L.EQ.IL) GO TO BDO 11,N= 1,NN(L +1)AJP=(NML(L+1)-I)+2*N-1BJP=AJP+1ABKA=0.0ABKB=0. 0DO 12,M=2,ML1LCALL YKPPM(WD,L,M,N,ALPHA,ML1,YRPP,YRPM)EZXII=EZ(L,M)/EX(L,M)/II(L,M,ALPHA)ABKA=ABKA+EZXII*YKPMABKB=ABKB+EZXII*YKPP

12 CONTINUEAB(IK,AJP)=-0.5/IIIKL*ABKAAB(IK,BJP)=0.5/IIIKL*ABKB

C(<11 CONTINUE8 CONTINUE7 CONTINUE

C<<<

154

CM(RETURNEND

*****************************************************SUBROUTINE EAKVKT(...)

* This subroutine computes EAK(L,M,N) and VKT(L,M,N)** defined in Chapter 4 of the thesis.*****************************************************

SUBROUTINE EAKVKT(IL,ML1,NN)COMPLEX*16 K(6,5,8),AX(6,5),AZ(6,5),EAK(6,5,8)

,CDTANH,Q(6,5,8),VKT(6,5,8)REAL*8 DX(6),Z(6,5),EX(E,5),EZ(6,5)DIMENSION ML1(6),NN(6)COMMON / DATE. / DX,ZCOMMON / DAT3 / EX,EZCOMMON / DAT4 / AX,AZCOMMON / DAT5 / K,QCOMMON / DAT21 / EAK, VKTDO 1, L=1,ILML1L=ML1(L)DO 2,M=2,ML1LDO 3,N=1,NN(L)VKT(L,M,N)=K(L,M,N)/K(L,2,N)*CDTANH(K(L,M,N)

*DX(L))EAK(L,M,N)=EX(L,M)*AX(L,M)**2*K(L,M,N)

3 CONTINUE2 CONTINUE1 CONTINUE

RETURNEND

****************************************************SUBROUTINE IILMN(...)

* This subroutine computes II(L,M,N).****************************************************

SUBROUTINE IILMN(IL,ML1,NN)COMPLEX*16 II(6,5,8),K(6,5,8),U(6,5,8),AX(6,5)

,AZ(6,5),DZZ,CIREAL*8 DX(6),Z(6,5)DIMENSION ML1(6),NN(6)COMMON / DATE / DX,ZCOMMON / DATA / AX, AZCOMMON / OATS / K,QCOMMON / DAT22 / II

CI=CMPLX(0.0,1.0)DO 1,L=1,ILDO 2,N=1,NN(L)II(L,2,N)=1.0

2 CONTINUE1 CONTINUEDO 3,L=1,ILML1L=ML1(L)

155

IF(ML1L.EQ.2) GO TO 3DO 4,M=3,ML1LDZZ=(Z(L,M-1)-Z(L,M))*AX(L,M)/AZ(L,M)DO 5,N=1,NN(L)DZZ=DZZ*K(L,M,N)II(L,M,N)=II(L,M-1,N)*(COS(DZZ)+CI*Q(L,M,N)

*SIN(DZZ))5 CONTINUE4 CONTINUE3 CONTINUE

RETURNEND

***************************************************SUBROUTINE IIIKD(...)

* This subroutine complements the subroutine* ABIJ(I,J).***************************************************

SUBROUTINE IIIKD(IL,MLI,NN)COMPLEX*16 K( 6,5,8),Q(6,5,8),RX(6,5),RZ(6,5)

a.

,CDSINH,II(6,5,8),QQ,KDX,ZZREAL*8 DX(6),Z(6,5),EX(6,5),EZ(6,5),Z1ZDIMENSION ML1(6),NN(6)COMMON / DAT2 / DX,ZCOMMON / DAT3 / EX,EZCOMMON / DAT4 / AX,AZCOMMON / DAIS / K,0COMMON / DAT22 / IICOMMON / DAT23 / IIIK,IIIDDO 1,L=1,ILML1L=ML1(L)DO 2,N=1,NN(L)IIIRDS=0.0DO 3,M=2,ML1LKAXZ=K(L,M,N)/AZ(L,M)*AX(L,M)Z1Z=Z(L,M-1)-Z(L,M)QQ=0(L,M,N)II1KDS=IIIKDS+EZ(L,M)/EX(L,M)/II(L,M,N)**2

& *ZZ(KAXZ,Z1Z,00)3 CONTINUE

IF(L.GT.1) GO TO 4IIIK(L,N)=IIIRDSGO TO 2

4 KDX=K(L,2,N)*DX(L)IIIK(L,N)=IIIKDS*CDSINH(KDX)IIID(L,N)=IIIKDS*EXP(-1.0*KDX)


RETURNEND

********************************************** ****

156

SUBROUTINE YKPPM(...)* This subroutine determines YKP(+) and YKP(-).

SUBROUTINE YKPPM(WD,L,M,N,ALPHA,ML1,YKPP,YKPM)COMPLEX*IE YKPP,YKPM,K(E,5,8),0(6,5,8),AX(6,5)

,II(6,5,8),EAK(6,5,8),VKT(E,5,8),K1,YKP,CDTANH,ZZLA,CDS,CDSINH,AZ(E,5),K2,01,02

REAL*8 DX(6),Z(6,5),WD,Z1,Z2,121INTEGER ALPHADIMENSION ML1(6)COMMON / DATE / DX,ZCOMMON / DAT4 / AX,AZCOMMON / DAT5 / K,0COMMON / DAT21 / EAK,VKTCOMMON / DAT22 / II

L1=L+1CDS=CDSINH(K(L1,2,N)*DX(L1))K1=K(LI,M,N)/AZ(L1,M)*AX(L1,M)Z1=WD-Z(L1,M)01=0(LI,M,N)K2=K(L,M,ALPHA)/AZ(L,M)*AX(L,M)Z2=WD-Z(L,M)Z21=WD-Z(L,M-1)02=0(L,M,ALPHA)YKP=EAK(L1,M,N)/EAK(L,2,ALPHA)

*ZZLA(K1,K2,ZI,Z2,Z21,01,02)/II(LI,M,N)/CDTANH(K(L1,M,N)*DX(LI))*CDS

YKPP=(1.0+VKT(L1,M,N))*YKPYKPM=(1.0-VKT(L1,M,N))*YKPIF(M.E0.ML1(L1)) GO TO 1K1=K2ZI=Z201=02M1=M+1K2=K(L1,M1,N)/AZ(L1,M1)*AX(L1,M1)Z2=WD-Z(L1,M1)Z21=WD-Z(L1,M)02=0(L1,M1,N)YKP=EAK(L1,M1,N)/EAK(L,2,ALPHA)

& *ZZLA(K1,K2,ZI,Z2,Z21,01,02)/II(L1,M1,N)/CDTANH(K(L1,M1,N)*DX(L1))*CDS

YKPP=YKPP+(1.0+VKT(LI,M1,N))*YKPYKPM=YKPM+(1.0-VKT(L1,M1,N))*YKP

1 RETURNEND

*****************************************************SUBROUTINE YKDMPM(...)

* This subroutine determines YKP(+), YKM(-), YDM(+),** and YDM(-).*****************************************************

157

SUBROUTINE YKDMPM (WD, L, M, N, ALPHA, ML I, YKMP, YKMM, YDMP, YDMM)

COMPLEX*16 YKMP, YKMM, YDMP, YDMM, K (6, 5, 8) , 0 (6, 5, 8), AZ (6, 5 ) , I I (6, 5, 8) , ERK (6, 5, 8)

, 01, 02, CCC, YKM, YDM, CDT, CDS, CDTRNHtf, AX (6, 5), VRT (6, 5, 8) , Kl, K2, CDSINH

, ZZLAREAL*8 DX (6) , Z (6, 5), WD, Z 1, Z2, Z21, EX (6, 5) , EZ (6, 5)

INTEGER RLPHRDIMENSION ML1 (6)COMMON / DAT2 / DX, ZCOMMON / DAT3 / EX, EZCOMMON / DAT4 / AX, AZCOMMON / DAT5 / K,QCOMMON / DAT21 / ERK, VKTCOMMON / DAT22 / I I

L1=L-1KI=K (L, M, ALPHA) /AZ (L, N)*AX (L, VI)

Z1=WD-Z (L, M)01=0 (L, M, ALPHA)K2=K (L1, M, N) /AZ (L1, M)*AX (LI, M)Z2=WD-Z (LI, M)Z21=WD-Z (L1, M-1)P2=0 (LI, M, N)CCC=EZ (L, M) /EX (L, M)*ZZLA (KI, K2, Z1, Z2, Z21

, 01, 02) / I I (L, M, ALPHA)

IF (M. EQ. ML1 (L) ) GO TO 1MI=M+1K1=K2Z1=Z201=02K2=K (L, Ml, ALPHA) /AZ (L, M1 )*AX (L, M1)Z2=WD-Z (L, M1)Z21=WD-Z (L, M)02=0 (L, MI, ALPHA)CCC=CCC+EZ (L, MI) /EX (L, MI )*ZZLA (K1, K2, Z1, Z2

, Z21, 01, 02) /II (L,M1, ALPHA)1 IF (L. ED. 2) GO TO 2

CDT=CDTANH (K (L1, M, N)*DX (L1) )CDS=CDSINH (K (L1, 2, N)*DX (L1) )YKM=CCC*EAK (LI, M, N) /EAR (L, 2, ALPHA)

/I I (L1, M, N) /CDT*CDS

YDM=CCC/ I I (L1, M, N) /CDT*CDSYKMP= (1. Q +VKT (L 1, M, N) ) *YKM

YKMM= ( I. O-VKT (L1, M, N) )*YKMYDMP= (K (L1, M, N) /K (L1, 2, N) +CDT )*YDM

YDMM= (K (LI, M, N) /K (L1, 2, N)-CDT)*YDMGO TO 3

2 YKMP=CCC*ERK (L1, M, N) /EAR (L, 2, ALPHA)YKMM=YKNIPYDMP=CCC

158

YDMM=YDMP3 RETURN

ENDSUBROUTINE EIGENDO(ML,NN,H,GH,DFZ,KWD12,KWDL2)

****************************************************SUBROUTINE EIGENDO(...)

* This subroutine is used to solve the dispersion ** equation.* The eigenvalues in the corresponding flow field ** (i.e. with the same wave period and water depth) ** in clear water are given as the initial guesses ** to solve for the eigenvalues in the flow field ** with the given madia properties but with zero* linear drag coefficient. Update the solutions as ** initial guesses and increase linear drag coeffi- ** cient with a small amount given by DFZ and solve ** the dispersion equation. Repeat the procedure* until the linear drag coefficients reach the* required values.* The subroutine calls another subroutine EIGEN* to solve the dispersion equation by the Secant ** method.

****************************************************COMPLEX*16 KWD12(8),AZL(5),OL(5,8),RWDL2(8)

,R(5),AXZI,GUESS,OLN(5),KH,AXL(5),CI,DZ(5)RERL *8 ZL(5),FZL(5),SZL(5),EXL(5),EZL(5),FZZ(5)

,GH,DFZ,FZZZ,TESTNA,TESTNB,FXL(5),SXL(5),FXX(5),H,FXXX,SMALL

COMMON / DAT12 / FXL,FZL,SXL,SZL,EXL,EZL,ZL& ,AXL,AZL,OLSMALL=1.0E-8CI=CMPLX(0.0,1.0)ML1=ML+1N1=1

C>>> Shooting solutions by starting from undampedC flow field:

1 DO 100, M=2,ML1FXX(M)=0.0FZZ(M)=0.0

100 CONTINUEC)) Assigning initial guesses:

DO 800,N=N1,NNKWDL2(N)=KWD12(N)

800 CONTINUEC(<

2 DO 200,M=2,ML1AXL(M)=SORT(-1.0/(SXL(M)+CI*FXX(M)))PaL(M)=SORT(-1.0/(SZL(M)+CI*FZZ(M)))

200 CONTINUEIF(ML.GT.1) GO TO 11R(2)=1.0

159

GO TO 1211 DO 300,M =2, ML

M1=M+1R(M)=(AZL(M1)/AZL(M))*(EZL(M1)/EZL(M))

*SORT(EXL(M)/EXL(M1))300 CONTINUE12 DO 400,M=1,ML

M1=M+1DZ(M)=(ZL(M)-ZL(M1))/H*SORT(EXL(2)/EXL(M1))

*(AXL(2)/AZL(M1))400 CONTINUE

AXZI=CI*AZL(2)*AXL(2)C)) Solving the NN eigenvalues of the toppest layer:

DO 600,N=N1,NNGUESS=KWDL2(N)CALL EIGEN(AXZI,GUESS,ML,ML1,GH,R,DZ,QLN,KM)KWDL2(N)=KHDO 500,M=2,ML1QL(M,N)=OLN(M)


C((MMM=2

C>) Testing if the linear drag coefficients areC the ones required:

3 IF(FZZ(MMM).LT.FZL(MMM)) GO TO 4IF(FXX(MMM).LT.FXL(MMM)) GO TO 4IF(MMM.EQ.ML1) GO TO 5MMM=MMM+1GO TO 3

C((4 DO 700,M=2,ML1

FXXX=FXX(M)FZZZ=FZZ(M)FXX(M)=FXXX+DFZFZZ(M)=FZZZ+DFZIF(FZZ(M).LT.FZL(M)) GO TO 701FZZ(M)=FZL(M)

701 IF(FXX(M).LT.FXL(M)) GO TO 700FXX(M)=FXL(M)

700 CONTINUEGO TO 2

CH(C))) Testing if all (KWDL2(N),N=1,NN) are differentc from each other. If not, halfing DFZ andC repeating above procedures.

5 IF(NN.EQ.1) GO TO 10NA=1NB=NA+1

6 TESTNA=ABS(MWDL2(NA))7 TESTNB=ABS(KWDL2(NB))

160

IF(ABS(TESTNA-TESTNB).LE.SMALL) GO TO 9NN1=NN-1IF(NA.EQ.NN1.AND.NB.EQ.NN) GO TO 10IF(NB.EQ.NN) GO TO 8NB=NB+1GO TO 7

8 NA=NA+1NB=NA+1GO TO 6

9 N1=NBDFZ=0.5*DFZGO TO 1

C(((10 RETURN

END***************************************************

SUBROUTINE EIGEN(...)* This subroutine solve the general dispersion* equation. The eigenvalues are complex numbers* in general.* The Secant method is applied in the subroutine. ****************************************************

SUBROUTINE EIGEN(A)(Z,GUESS,ML,ML1,GH,RLM,DZLM,QN,Y)

COMPLEX*16 DZLM(5).RLM(5),QN(5),Q0(5),YO,YN,Y,GUESS,CDTAN,CCTAN,DY,OFO,OFN,AXZ,CI

REAL*8 GH,S0S1,S0S2,DIVERGECI=CMPLX(0.0,1.0)SOS1=1.0E-10SOS2=1.0E-10DIVERGE=1.0E+7YO=GUESSYN=GUESS*(1.0+1.0E-7)

*** Note: The index has been shifted such that*** QM is named as 0(2), Q(ML) as Q(ML+1)*** Array(m)----Array(m+1), ETC.*** ML1=ML+1

00(MLI)=0.0ON(ML1)=Q0(ML1)IF(ML.EQ.1) GO TO 30MLM1=ML-1DO 20 I=1,MLM1M=ML1-IM1=M+1CCTAN=CI*CDTAN(DZLM(M)*Y0)00(M)=RLM(M)*(CCTAN+00(M1))/(1.0+CCTAN*Q0(M1))

20 CONTINUE30 CCTAN=CI*CDTAN(DZLM(1)*Y0)

OFO=GH+AXZ*Y0*(CCTAN+00(2))/(1.0+CCTAN*00(2))1 IF(ML.EQ.1) GO TO 40DO 10 I=1,MLM1

161

M=ML1-IM1=M+1CCTAN=CI*CDTAN(DZLM(M)*YN)ON(M)=RLM(M)*(CCTAN+ON(M1))/(1.0+CCTAN*ON(M1))

10 CONTINUE40 CCTAN=CI*CDTAN(DZLM(1)*YN)

OFN=GH+AXZ*YN*(CCTAN+ON(2))/(1.0+CCTAN*CON(2))DY=OFN*(YN-Y0)/(UFN-OF0)IF(ABS(QFN).LE.SOS1) GO TO 3IF(ABS(DY).LE.S0S2) GO TO 3IF(ABS(DY).GE.DIVERGE) GO TO 6YO=YNYN=YN-DY0F0=OFNGO TO 1

3 Y=YNRETURN

6 WRITE(*,7)7 FORMAT(1X,' DIVERGENT ""111)44 STOP

END***********************************************

FUNCTION CDTAN(Z)COMPLEX*16 Z,CDTAN,CDS,CDCCDS=SIN(7)CDC=COS(Z)CDTAN=CDS/CDCRETURNEND

*****************************************************SUBROUTINE KHJ1(...)

* This subroutine solves the dispersion equation of ** propagating wave in clear water by the Newton* method.*****************************************************

SUBROUTINE KHJ1(GH,KH1)REAL*8 KH1,GH,SMALL,PI,X,DATAXSMALL =1. OE -5PI=4.0*ATAN(1.0)X=GHIF(X.GE.3.5*PI) GO TO 2

1 DATRX=(GH-X*TANH(X))/(X/COSH(X)**2+TANH(X))IF(ABS(DATAX).LE.SMALL) GO TO 2X=X+DATRXGO TO 1

2 KH1=XRETURNEND

*****************************************************SUBROUTINE KHJJ(...)

* This subroutine solves the dispersion equation of *

162

* the evanescent wave modes in clear water by the ** Newton method.*****************************************************

SUBROUTINE KHJJ(J,GH,KHJ)REAL*6 KHJ,GH,SMALL,PI,X,DATAXSMALL=1.0E-5PI=4.0*ATAN(1.0)X=(2*J-3)*PI/2.0+0.01

1 DATAX=(GH+X*TAN(X))/(X/COS(X)**2+TAN(X))IF(ABS(DATAX).LE.SMALL) GO TO 2X=X-DATAXGO TO 1

2 KHJ=XRETURNENDSUBROUTINE MATRIXC(N,D)

****************************************************SUBROUTINE MRTRIXC(...)

* This subroutine solves a matrix equation by a* method modified from Gaussian elimination method.*****************************************************

COMPLEX*16 X(68),B(68),A(86,86),D,C(88),BB,XXDIMENSION IP(88)COMMON / DAT6 / ACOMMON / DAT61 / B,XCOMMON / DAT62 / IPCALL FACTOR(N,D)IF(ABS(D).EO.0.0) GO TO 999DO 1,I=1,NC(I)=B(I)

1 CONTINUEDO 2,I=1,NIC=IP(I)B(I)=C(IC)

2 CONTINUEB(1)=B(1)/A(1,1)DO 3,I=2,NBB=B(I)I1=I-1DO 4,K=1,I1BB=BB-A(I,K)*B(K)

4 CONTINUEB(I)=BB/A(I,I)

3 CONTINUEX(N)=B(N)N1=N-1DO 5,I=N1,1,-1XX=B(I)II1=I+1DO 6,K=II1,NXX=XX-A(I.K)*X(K)

163

6 CONTINUEX(I)=XX

5 CONTINUE999 RETURN

END************************************* ******

SUBROUTINE FACTOR(N,D)REAL*13 DI,DTESTDIMENSION IP(88)COMMON / DATE / ACOMMON / DAT62 / IPDTEST=1.0E-10DO 1,I=1,NIP(I)=I

1 CONTINUEDI=1.0II=1CALL PIVOT(II,N,DI)IF(ABS(A(II,II)).LE.DTEST) GO TO 999II1=II+1DO 2,J=II1,NA(II,J)=A(II,J)/A(II,II)

2 CONTINUEII=2

102 DO 3,I=II,NL=A(I,II)IIM1=II-1DO 4,K=1,IIM1L=L-A(I,K)*A(K,II)

4 CONTINUEA(I,II)=L

3 CONTINUEIF(II.EGLN) GO TO 101CALL PIVOT(II,N,DI)IF(ABS(A(II,II)).LE.DTEST) GO TO 999II11=II+1DO 5,J=II11,NU=A(ii,j)1121=11-1DO 6,K=1,1121U=U-A(II,K)*A(K,J)

6 CONTINUEA(II,J)=U/R(II,II)

5 CONTINUEII=II+1GO TO 102

101 D=DIDO 7,I=1,ND=D*121(I,I)

7 CONTINUEGO TO 103

164

999 D=0.0WRITE (*,*) II

103 RETURNEND

**************************************SUBROUTINE PiIVDT ( I I, N, DI)COMPLEX*16 A (88,88), RJ, CRERL *8 DIDIMENSION IP (88)COMMON / DATE, / ACOMMON / DRT62 / IPIM=I I

RJ=A (II, I I)

1121=11+1DO 1, K=I IP1, NIF (ABS (AJ) . GE. ABS (R (K, II ) ) ) GO TO 1AJ=R (K, II )IM=K

1 CONTINUEIF ( I M. EQ. II) GO TO 100DI=-1.0*DIDO 2, J=1, NC=A ( I I , 3)

R(II, J) =A ( IM, J.)

A(IM,J)=CCONTINUEIC=IP (I I)

IP(II)=IP(IM)IP (IM)=IC

100 RETURNENDSUBROUTINE UW (H, L, M, NN)

****************************************************SUBROUTINE UW (... )

* This subroutine calculates the absolute values of** z), W(x,z), U"a(x, z), (x, z), U'3(x,z), and** W-3 (x, z) at the grid points given by (MML(L)* NNLM (L, M) ).****************************************************

REAL*8 X (6), DX (6), Z (6,5), XV (5) , ZV (10), UA, WAUR1 (5,10) , WA1 (5,10) , LIA2 (5,10), WA2 (5,10)

, UR3 (5,10), WA3 (5,10), H, DXV, DZV, XVX, ZVHZDIMENSION MML (6), NNLM (6,5), NN (6)COMMON / DAT1 / XCOMMON / DAT2 / DX, ZCOMMON / DAT10 / MML, NNLMCOMMON / DAT11 / UP11, WAl, UR2, WAG', UA3, WR3DXV=2.0*DX (L) /MML (L)M1=M-1DZV= (Z (L,N1 ) -Z (L, ) /NNLM (L, M)MMLL1=MML (L)

165

NNLMLM1=NNLM(L,M)+1XV(1)=X(L)-DX(L)DO 1,I=2,MMLL1XV(I)=XV(1)+DXV*(I-1)

1 CONTINUEZV(1)=-1.0*H+Z(L,M)DO 2,J=2,NNLMLM1ZV(J)=ZV(1)+DZV*(J-1)

2 CONTINUEDO 3,I=1,MMLL1XVX=XV(I)-X(L)DO 4,J=1,NNLMLM1ZVHZ=ZV(J)+H-Z(L,M)CALL UWXZ(L,M,NN,XVX,ZVHZ,UA,WA)UAl(I,J)=UAWAl(I,J)=WAUA2(I,J)=UA1(I,J)**2WA2(I,J)=WA1(I,J)**2UA3(1,3)=UA1(I,J)*UA2(I,3)WA3(I,J)=WA1(I,J)*WA2(I,J)


RETURNEND

*************************************************SUBROUTINE UWXZ(L,M,NN,XVX,ZVHZ,UA,WA)

****************************************************SUBROUTINE UWXZ(...)

* This subroutine calculates the absolute value of ** U(x,z) and W(x,z) at the given position (x,z). ** XVX=x-X(L)* ZVHZ=z+h-Z(L,M)****************************************************

COMPLEX*16 CI,AX(6,5),AZ(6,5),K(E,5,8),0(6,5,8),B(6,5,8),USUM,WSUM,KAX,KADX,EXPP,XLMN,XPLMN,KAZ,RLMNZICDSR,SINR,A(6,5,8),EXPM,AKX,BKX,ZZLMN,ZPLMN

REAL *8 XVX,ZVHZ,UA,WADIMENSION NN(S)COMMON / DAT4 / AX,AZCOMMON / DATS / KJ?COMMON / DAT7 / A,BCI=CMPLX(0.0,1.0)USUM =0. 0WSUM=0.0DO 5,N=1,NN(L)KAX=K(L,M,N)KADX=KAX*XVXEXPP=EXP(KADX)EXPM=EXP(-1.0*KADX)AKX=A(L,M,N)*EXPP

166

BKX=B(L,M,N)*EXPMXLMN=AKX+BKXXPLMN=KAX*(AKX-BKX)KAZ=K(L,M,N)/AZ(L,M)*AX(L,M)RLMNZ=KAZ*ZVHZCDSR=COS(RLMNZ)SINR=SIN(RLMNZ)ZZLMN=CDSR+CI*D(L,M,N)*SINRZPLMN=KAZ*(-1.0*SINR+CI*12(L,M,N)*CDSR)USUM=USUM+XPLMN*ZZLMNWSUM=WSUM+XLMN*ZPLMN

5 CONTINUEUA=ABS(USUM*AX(L,M)**2)WA=ABS(WSUM*AZ(L,M)**2)RETURNEND

****************************************************SUBROUTINE PPD(WD,P11,IL,NNN)

***************************************************** SUBROUTINE PPD(...)* This subroutine calculates the relative amplitude** of dynamic pressure at the given grid points:* (MML(L)+1,NNLMLM(L,M)+1).****************************************************

REAL*8 X(6),DX(6),Z(6,5),XP(5),ZP(10),XPX,ZPZ,P(6,5,10),DXP,DZP,P11,WD,PDLM

DIMENSION MML(6),NNLM(6,5),NNN(6)COMMON / DAT1 / XCOMMON / DAT2 / DX,ZCOMMON / DAT10 / MML,NNLMCOMMON / DAT13 / PDO 1,L=2,ILM=3M1=M-1DXP=2.0*DX(L)/MML(L)DZP=(Z(L,M1)-Z(L,M))/NNLM(L,M)MMLL1=MML(L)+1NNLMLM1=NNLM(L,M)+1XP(1)=X(L)-DX(L)DO 2,I=2,MMLL1XP(I)=XP(1)+DXP*(I-1)

2 CONTINUEZP(1)=-1.0*WD+Z(L,M)DO 3,J=2,NNLMLM1ZP(J)=ZP(1)+DZP*(3-1)

3 CONTINUEDO 4,I=1,MMLL1XPX=XP(I)-X(L)DO 5,J= 1,NNLMLMIZPZ=ZP(J)+WD-Z(L,M)CALL PD(L.M,NNN,XPX,ZPZ,PDLM)

167

P(L,I,J)=PDLM/P115 CONTINUE4 CONTINUE1 CONTINUE

RETURNEND

****************************************************SUBROUTINE PD(L,M,NN,PX,PZ,PDLM)

***************************************************** This subroutine calculates the dynamic pressure ** at (x,z): PLM(x,z)* PX=x-X(L)* PZ=z+h-Z(L,M)****************************************************

COMPLEX*16 CI,AX(6,5),AZ(6,5),K(6,5,6),O(6,5,8),A(6,5,6),B(6,5,8),PPP,KX,KZ

REAL*6 PX,PZ,PDLMDIMENSION NN(6)COMMON / DAT4 / AX,AZCOMMON / DAT5 / K,QCOMMON / DAT7 / A,BCI=CMPLX(0.0,1.0)PPP =C). 0

DO 1,N=1,NN(L)KX=K(L,M,N)*PXKZ=K(L,M,N)/(AZ(L,M)/AX(L,M))*PZPPP=PPP+(A(L,M,N)*EXP(KX)+B(L,M,N)*EXP(-1.0

& *KX))*(COS(KZ)+CI*O(L,M,N)*SIN(KZ))1 CONTINUE

PDLM=ABS(PPP)RETURNEND

******************************************************SUBROUTINE SITR(HX,HY,MM,NN,F,SI)

******************************************************SUBROUTINE SITR(...)

* This subroutine numerically calculates a surface ** integration by the Trapezoidal Rule.******************************************************

REAL*8 F(5,10),SI,HX,HY,HXY

HXY=HX*HYNN1=NN+1MM1=MM+1si=0.25*HXY*(F(1,1)+F(1,NN1)+F(MM1,1)+F(MM1,NNU)DO 1,M=2,MMSI=SI+0.5*HXY*(F(M,1)+F(M,NN1))

1 CONTINUEDO 2,N=2,NN5I=SI+0.5*HXY*(F(1,N)+F(MM1,N))

2 CONTINUE

168

DO 3,M=2,MMDO 4,N=2,NNSI=SI+HXY*F(M,N)


RETURNEND

*****************************************************FUNCTION ZZLA(...)

* The function is defined as < ZLMN(z) *ZPQR(z) ) for** L is not equal to P.

K1=K(L,M,N)/(AZ(L,M)/AX(L,M))Z1=h-Z(L,M)Q1= Q(L,M,N)K2=k(P,O,R)/(AZ(P,O)/AX(P,O))Z2=h-Z(P,Q)Z21=h-Z(P,O-1)02=0(P,O,R)

*****************************************************FUNCTION ZZLA(K1,K2,21,Z2,Z21,01,02)COMPLEX*18 K1,K2,Q1,02,THM,THP,DTHM,DTHP,ATHM

& ,CI,ATHP,DZ,ZZLAREAL*8 Z1,Z2,Z21,ZM,ZP,SMALLCI=CMPLX(0.0,1.0)SMALL=1.0E-5ZM=Z1-Z21ZP=(Z1+Z21)/2.0THM=K1-K2THP=Kl+K2DTHM=Z1*K1-Z2*K2DTHP=Z1*K1+22*K2DZ=(Z2-Z21)*K2IF(ABS(THM).LE.SMALL) GO TO 1IF(ABS(THP).LE.SMALL) GO TO 2ATHM=DTHM-ZP*THMATHP=DTHP-ZP*THPZZLA=1.0/THP*SIN(ZM/2.0*THP)*((1.0+01*O2)*COS

(ATHP)+CI*(01+02)*SIN(ATHP))+1.0/THM*SIN(ZM/2.0*THM)*((1.0-01*02)*COS

& (ATHP)+CI*(01-02)*SIN(ATHM))GO TO 3

1 ZZLA=1.0/THP*SIN(ZM/2.0*THP)*((1.0+01*02)*COS(DZ)+CI*(U1+02)*SIN(DZ))

+0.5*((1.0-01*02)*COS(DTHM)+CI*(01-02)*SIN(DTHM))*ZM

GO TO 32 ZZLA=1.0/THM*SIN(ZM/2.0*THM)*((1.0-01*02) COS

(DZ)-CI*(0.1-Q2)*SIN(DZ))& +0.5*((1.0+O1 *O2)*COS(DTHP)+CI*(01+02)

*SIN(DTHP)) *ZM3 RETURN

169

END****************************************************

FUNCTION ZZ(...)* The function is defined as ( ZLMN(z)*ZPQR(z) ) ** for L=P, M=0, and N=R.

KAXZ=K(L,M,N)/(AZ(L,M)/AX(L,M))Z1Z=Z(L,M-1)-Z(L,M)

****************************************************FUNCTION ZZ(KAXZ,Z1Z,Q)COMPLEX*16 KAXZ,Q,KZZZ,CIREAL*8 Z1ZCI=CMPLX(0.0,1.0)KZ=KAXZ*Z1ZZZ=0.5/KAXZ*SIN(KZ)*((1.0+0**2)*COS(KZ)+2.0*CI

& *Q*SIN(KZ))+Z1Z/2.0*(1.0-0**2)RETURNEND

****************************************************FUNCTION CDSINH(X)COMPLEX*16 CI, CDSINH, XCI=CMPLX(0.0,1.0)CDSINH=SIN(CI*X)/CIRETURNEND

****************************************************FUNCTION CDTANH(X)COMPLEX*16 CI,CDTANH,XCI=CMPLX(0.0,1.0)CDTANH=SIN(CI*X)/COS(CI*X)/CIRETURNENDSUBROUTINE FIL(IL,ML1,NN,NNN)

****************************************************SUBROUTINE FIL(...)

* This subroutine excludes the waves that are* amplifying while they are propagating, and re- ** defines the order of the eigervalues.****************************************************

COMPLEX*16 K(6,5,8),Q(6,5,8)REAL*8 KRKIDIMENSION NT(8),NNN(E),ML1(6)COMMON / DAT5 / K,0NNN(1)=NNDO 1,L=2,ILDO 2,N=1,NNNT(N)=N

2 CONTINUEML1L=ML1(L)DO 3,M=2,ML1LJ=1DO 4,N=1,NN

170

IF (NT (N) N) GO TO 101GO TO 4

101 KRKI=DREAL (K (L, M, N) ) *DIMAS ( ( L, M, N) )

IF (K RR I. GT. O. 0) GO TO 102NTN=JJ=J+1GO TO 4

102 NT (N)=NN+14 CONTINUE3 CONTINUE

NNN (L)=NTNDO 5, r1=2, IlL1L

J=1DO 6, N=1, NNIF (NT (N) ED. NN+1 ) GO TO 6K (L, J)=K (L, M, NT (N) )

C(L, M, J)=CI(L,M, NT (N) )

J=J+16 CONTINUE5 CONTINUE1 CONTINUE

RETURNEND

Wave interaction with permeable structures - Oregon State ...

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